Integral

Integral

Overview


Integration is an important concept in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and, together with its inverse, differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

, is one of the two main operations in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. Given a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 ƒ of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 x and an interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [ab] of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

, the definite integral


is defined informally to be the net signed area of the region in the xy-plane bounded by the graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 of ƒ, the x-axis, and the vertical lines x = a and x = b.

The term integral may also refer to the notion of antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

, a function F whose derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 is the given function ƒ.
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Encyclopedia


Integration is an important concept in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and, together with its inverse, differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

, is one of the two main operations in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. Given a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 ƒ of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 x and an interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [ab] of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

, the definite integral


is defined informally to be the net signed area of the region in the xy-plane bounded by the graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

 of ƒ, the x-axis, and the vertical lines x = a and x = b.

The term integral may also refer to the notion of antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

, a function F whose derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 is the given function ƒ. In this case, it is called an indefinite integral and is written:
The integrals discussed in this article are termed definite integrals.

The principles of integration were formulated independently by Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 and Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in the late 17th century. Through the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

, which they independently developed, integration is connected with differentiation
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

: if ƒ is a continuous real-valued function defined on a closed interval [ab], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by


Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

. The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

 width. A rigorous mathematical definition of the integral was given by Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

 is defined for functions of two or three variables, and the interval of integration [ab] is replaced by a certain curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

 connecting two points on the plane or in the space. In a surface integral
Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...

, the curve is replaced by a piece of a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 in the three-dimensional space.
Integrals of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, and they play an important role in the formulation of many physical laws, notably those of electrodynamics
Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

, developed by Henri Lebesgue
Henri Lebesgue
Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...

.

Pre-calculus integration


Integration can be traced as far back as ancient Egypt
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...

 ca. 1800 BC, with the Moscow Mathematical Papyrus
Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...

 demonstrating knowledge of a formula for the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 of a pyramidal frustum
Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

. The first documented systematic technique capable of determining integrals is the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

 of the ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

 astronomer Eudoxus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...

 (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

 in the 3rd century BC and used to calculate areas for parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

s and an approximation to the area of a circle. Similar methods were independently developed in China around the 3rd century AD by Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

 and Zu Geng to find the volume of a sphere. That same century, the Indian mathematician
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

 Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...

 used a similar method in order to find the volume of a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

.

The next major step in integral calculus came from the Abbasid Caliphate when the 11th century mathematician
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...

 Ibn al-Haytham (known as Alhazen in Europe) devised what is now known as "Alhazen's problem", which leads to an equation of the fourth degree, in his Book of Optics
Book of Optics
The Book of Optics ; ; Latin: De Aspectibus or Opticae Thesaurus: Alhazeni Arabis; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Muslim scholar Alhazen .-See also:* Science in medieval Islam...

. While solving this problem, he performed an integration in order to find the volume of a paraboloid
Paraboloid
In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

. Using mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

, he was able to generalize his result for the integrals of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree. Some ideas of integral calculus are also found in the Siddhanta Shiromani, a 12th century astronomy text by Indian mathematician Bhāskara II.

The next significant advances in integral calculus did not begin to appear until the 16th century. At this time the work of Cavalieri
Bonaventura Cavalieri
Bonaventura Francesco Cavalieri was an Italian mathematician. He is known for his work on the problems of optics and motion, work on the precursors of infinitesimal calculus, and the introduction of logarithms to Italy...

 with his method of indivisibles
Cavalieri's principle
In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...

, and work by Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of xn up to degree n = 9 in Cavalieri's quadrature formula. Further steps were made in the early 17th century by Barrow
Isaac Barrow
Isaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...

 and Torricelli
Evangelista Torricelli
Evangelista Torricelli was an Italian physicist and mathematician, best known for his invention of the barometer.-Biography:Evangelista Torricelli was born in Faenza, part of the Papal States...

, who provided the first hints of a connection between integration and differentiation
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

. Barrow provided the first proof of the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.

At around the same time, there was also a great deal of work being done by Japanese mathematicians
Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo Period . The term wasan, from wa and san , was coined in the 1870s and employed to distinguish native Japanese mathematics theory from Western mathematics .In the history of mathematics, the development of wasan...

, particularly by Seki Kōwa. He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

.

Newton and Leibniz


The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

 by Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 and Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, whose notation for integrals is drawn directly from the work of Leibniz.

Formalizing integrals


While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley
George Berkeley
George Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...

 memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

 and was given a suitable foundation by Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

 in the first half of the 19th century. Integration was first rigorously formalized, using limits, by Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

. Although all bounded piecewise continuous functions are Riemann integrable on a bounded interval, subsequently more general functions were considered – particularly in the context of Fourier analysis – to which Riemann's definition does not apply, and Lebesgue
Henri Lebesgue
Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...

 formulated a different definition of integral, founded in measure theory
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 (a subfield of real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...

). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed.

Notation


Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with or , which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

The modern notation for the indefinite integral was introduced by Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 in 1675 . He adapted the integral symbol, , from an letter ſ (long s
Long s
The long, medial or descending s is a form of the minuscule letter s formerly used where s occurred in the middle or at the beginning of a word, for example "ſinfulneſs" . The modern letterform was called the terminal, round, or short s.-History:The long s is derived from the old Roman cursive...

), standing for summa (written as ſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

 in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 .

Terminology and notation


If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. The region over which a function is being integrated is called the domain of integration. Usually this domain will be an interval, in which case it is enough to give the limits of that interval, which are called the limits of integration. If the integral does not have a domain of integration, it is considered indefinite (one with a domain is considered definite). In general, the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense (such as a sample space in probability theory).

The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted by

The ∫ sign represents integration; a and b are the lower limit and upper limit, respectively, of integration, defining the domain of integration; f is the integrand, to be evaluated as x varies over the interval [a,b]; and dx is the variable of integration. In correct mathematical typography, the dx is separated from the integrand by a space (as shown). Some authors use an upright d (that is, dx instead of dx).

The variable of integration dx has different interpretations depending on the theory being used. For example, it can be seen as strictly a notation indicating that x is a dummy variable of integration, as a reflection of the weights in the Riemann sum
Riemann sum
In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....

, a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 (in Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

 and its extensions), an infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

 (in non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

) or as an independent mathematical quantity: a differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

. More complicated cases may vary the notation slightly.

In the modern Arabic mathematical notation
Modern Arabic mathematical notation
The designation modern Arabic mathematical notation is used for a mathematical notation based on the Arabic script that is widely used in the Arab world, especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it...

, which aims at pre-university levels of education in the Arab world and is written from right to left, a reflected integral symbol is used .

Introduction


Integrals appear in many practical situations. If a swimming pool is rectangular with a flat bottom, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering
Precision engineering
Precision engineering is a subdiscipline of electrical engineering, electronics engineering, mechanical engineering, and optical engineering concerned with designing machines, fixtures, and other structures that have exceptionally low tolerances, are repeatable, and are stable over time...

 (of any discipline) requires exact and rigorous values for these elements.


To start off, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = √x. We ask:
What is the area under the function f, in the interval from 0 to 1?

and call this (yet unknown) area the integral of f. The notation for this integral will be

As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 15, 25, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √15, √25, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely


Notice that we are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an approximate value for the area of 0.6203, which is too small. The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely many fine, or infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

steps.

As for the actual calculation of integrals, the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

, due to Newton and Leibniz, is the fundamental link between the operations of differentiating
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 and integrating. Applied to the square root curve, f(x) = x1/2, it says to look at the antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

 F(x) = 23x3/2, and simply take F(1) − F(0), where 0 and 1 are the boundaries of the interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [0,1]. So the exact value of the area under the curve is computed formally as

(This is a case of a general rule, that for f(x) = xq, with q ≠ −1, the related function, the so-called antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

 is F(x) = (xq+1)/(q + 1).)

The notation
conceives the integral as a weighted sum, denoted by the elongated s, of function values, f(x), multiplied by infinitesimal step widths, the so-called differentials, denoted by dx. The multiplication sign is usually omitted.

Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

 of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation
refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. Here A denotes the region of integration.

Differential geometry, with its "calculus on manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s", gives the familiar notation yet another interpretation. Now f(x) and dx become a differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

, ω = f(x) dx, a new differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

 d, known as the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

 is introduced, and the fundamental theorem becomes the more general Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

,
from which Green's theorem
Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...

, the divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

, and the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

 follow.

More recently, infinitesimals have reappeared with rigor, through modern innovations such as non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

. Not only do these methods vindicate the intuitions of the pioneers; they also lead to new mathematics.

Although there are differences between these conceptions of integral, there is considerable overlap. Thus, the area of the surface of the oval swimming pool can be handled as a geometric ellipse, a sum of infinitesimals, a Riemann integral, a Lebesgue integral, or as a manifold with a differential form. The calculated result will be the same for all.

Formal definitions


There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.

Riemann integral




The Riemann integral is defined in terms of Riemann sum
Riemann sum
In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....

s of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 of the real line; then a tagged partition of [a,b] is a finite sequence



This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as
thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δi = xixi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1…n Δi. The Riemann integral of a function f over the interval [a,b] is equal to S if:
For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have

When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum
Darboux integral
In real analysis, a branch of mathematics, the Darboux integral or Darboux sum is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of...

, suggesting the close connection between the Riemann integral and the Darboux integral
Darboux integral
In real analysis, a branch of mathematics, the Darboux integral or Darboux sum is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of...

.

Lebesgue integral


The Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory). For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions are integrable . The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum.

The definition of the Lebesgue integral thus begins with a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

, μ. In the simplest case, the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 μ(A) of an interval A = [a,b] is its width, ba, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

To exploit this flexibility, Lebesgue integrals reverse the approach to the weighted sum. As puts it, "To compute the Riemann integral of f, one partitions the domain [a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f".

One common approach first defines the integral of the indicator function of a measurable set
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 A by:.
This extends by linearity to a measurable simple function
Simple function
In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line which attains only a finite number of values...

 s, which attains only a finite number, n, of distinct non-negative values:
(where the image of Ai under the simple function s is the constant value ai). Thus if E is a measurable set one defines
Then for any non-negative measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

 f one defines
that is, the integral of f is set to be the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

 of all the integrals of simple functions that are less than or equal to f.
A general measurable function f, is split into its positive and negative values by defining
Finally, f is Lebesgue integrable if
and then the integral is defined by

When the measure space on which the functions are defined is also a locally compact
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...

 topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...

s, of which the Lebesgue measure is an example) and integral with respect to them can be defined differently, starting from the integrals of continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

s with compact support. More precisely, the compactly supported functions form a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 that carries a natural topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, and a (Radon) measure can be defined as any continuous linear functional on this space; the value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by and a certain number of other authors. For details see Radon measures.

Other integrals


Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:
  • The Riemann–Stieltjes integral, an extension of the Riemann integral.
  • The Lebesgue-Stieltjes integral
    Lebesgue-Stieltjes integration
    In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework...

    , further developed by Johann Radon
    Johann Radon
    Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations .- Life :...

    , which generalizes the Riemann–Stieltjes and Lebesgue integrals
    Lebesgue integration
    In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

    .
  • The Daniell integral
    Daniell integral
    In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced...

    , which subsumes the Lebesgue integral
    Lebesgue integration
    In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

     and Lebesgue-Stieltjes integral
    Lebesgue-Stieltjes integration
    In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework...

     without the dependence on measure
    Measure (mathematics)
    In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

    s.
  • The Henstock-Kurzweil integral
    Henstock-Kurzweil integral
    In mathematics, the Henstock–Kurzweil integral, also known as the Denjoy integral and the Perron integral, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral which in some situations is more useful than the Lebesgue integral.This integral...

    , variously defined by Arnaud Denjoy
    Arnaud Denjoy
    Arnaud Denjoy was a French mathematician.Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives...

    , Oskar Perron
    Oskar Perron
    Oskar Perron was a German mathematician.He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951...

    , and (most elegantly, as the gauge integral) Jaroslav Kurzweil
    Jaroslav Kurzweil
    Jaroslav Kurzweil is a Czech mathematician. He is a specialist in ordinary differential equations and defined the Henstock–Kurzweil integral in terms of Riemann sums...

    , and developed by Ralph Henstock
    Ralph Henstock
    Ralph Henstock was an English mathematician and author. As an Integration theorist, he is notable for Henstock–Kurzweil integral...

    .
  • The Itō integral
    Ito calculus
    Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations....

     and Stratonovich integral
    Stratonovich integral
    In stochastic processes, the Stratonovich integral is a stochastic integral, the most common alternative to the Itō integral...

    , which define integration with respect to semimartingale
    Semimartingale
    In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

    s such as Brownian motion
    Wiener process
    In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

    .


  • The Young integral, which is a kind of Riemann-Stieltjes integral with respect to certain functions of unbounded variation
    Bounded variation
    In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...

    .
  • The rough path integral defined for functions equipped with some additional "rough path" structure, generalizing stochastic integration against both semimartingale
    Semimartingale
    In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

    s and processes such as the fractional Brownian motion.

Linearity

  • The collection of Riemann integrable functions on a closed interval [a, b] forms a vector space
    Vector space
    A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

     under the operations of pointwise addition and multiplication by a scalar, and the operation of integration

is a linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

 on this vector space. Thus, firstly, the collection of integrable functions is closed under taking linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

s; and, secondly, the integral of a linear combination is the linear combination of the integrals,



  • Similarly, the set of real
    Real number
    In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

    -valued Lebesgue integrable functions on a given measure space
    Measure (mathematics)
    In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

     E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral


is a linear functional on this vector space, so that


  • More generally, consider the vector space of all measurable function
    Measurable function
    In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

    s on a measure space (E,μ), taking values in a locally compact
    Locally compact space
    In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...

     complete topological vector space
    Topological vector space
    In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

     V over a locally compact topological field
    Topological ring
    In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...

     K, f : EV. Then one may define an abstract integration map assigning to each function f an element of V or the symbol ,
that is compatible with linear combinations. In this situation the linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R, C, or a finite extension of the field Qp of p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s, and V is a finite-dimensional vector space over K, and when K=C and V is a complex Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

.


Linearity, together with some natural continuity properties and normalisation for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell
Daniell integral
In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced...

 for the case of real-valued functions on a set X, generalized by Nicolas Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...

 to functions with values in a locally compact topological vector space. See for an axiomatic characterisation of the integral.

Inequalities for integrals


A number of general inequalities hold for Riemann-integrable functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 defined on a closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 and bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

 interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell).
  • Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded
    Bounded function
    In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...

     on that interval. Thus there are real number
    Real number
    In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

    s m and M so that mf (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(ba) and M(ba), it follows that

  • Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus
This is a generalization of the above inequalities, as M(ba) is the integral of the constant function with value M over [a, b].
In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if f(x) < g(x) for each x in [a, b], then

  • Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then

  • Products and absolute values of functions. If f and g are two functions then we may consider their pointwise product
    Pointwise product
    The pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain...

    s and powers, and absolute value
    Absolute value
    In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

    s:
If f is Riemann-integrable on [a, b] then the same is true for |f|, and
Moreover, if f and g are both Riemann-integrable then f 2, g 2, and fg are also Riemann-integrable, and
This inequality, known as the Cauchy–Schwarz inequality
Cauchy–Schwarz inequality
In mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...

, plays a prominent role in Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 theory, where the left hand side is interpreted as the inner product
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

 of two square-integrable
Square-integrable function
In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...

 functions f and g on the interval [a, b].

  • Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemann-integrable functions. Then the functions |f|p and |g|q are also integrable and the following Hölder's inequality
    Hölder's inequality
    In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....

     holds:
    For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.

    • Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then |f|p, |g|p and |f + g|p are also Riemann integrable and the following Minkowski inequality
      Minkowski inequality
      In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp...

       holds:
      An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces
      Lp space
      In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

      .

      Conventions


      In this section f is a real-
      Real number
      In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

      valued Riemann-integrable function
      Function (mathematics)
      In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

      . The integral
      over an interval [a, b] is defined if a < b. This means that the upper and lower sums of the function f are evaluated on a partition a = x0x1 ≤ . . . ≤ xn = b whose values xi are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [xi , xi +1] where an interval with a higher index lies to the right of one with a lower index. The values a and b, the end-points of the interval
      Interval (mathematics)
      In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

      , are called the limits of integration
      Limits of integration
      In calculus and mathematical analysis the limits of integration of the integral \int_a^b f \, dx of a Riemann integrable function f defined on a closed and bounded [interval] are the real numbers a and b.-Improper integrals:...

       of f. Integrals can also be defined if a > b:
      • Reversing limits of integration. If a > b then define

      This, with a = b, implies:
      • Integrals over intervals of length zero. If a is a real number
        Real number
        In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

         then


      The first convention is necessary in consideration of taking integrals over subintervals of [a, b]; the second says that an integral taken over a degenerate interval, or a point
      Point (geometry)
      In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

      , should be zero
      0 (number)
      0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

      . One reason for the first convention is that the integrability of f on an interval [a, b] implies that f is integrable on any subinterval [c, d], but in particular integrals have the property that:
      • Additivity of integration on intervals. If c is any element of [a, b], then

      With the first convention the resulting relation

      is then well-defined for any cyclic permutation of a, b, and c.

      Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed of differential forms on oriented manifolds
      Orientability
      In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

       only. If M is such an oriented m-dimensional manifold, and M is the same manifold with opposed orientation and ω is an m-form, then one has:

      These conventions correspond to interpreting the integrand as a differential form, integrated over a chain
      Chain (algebraic topology)
      In algebraic topology, a simplicial k-chainis a formal linear combination of k-simplices.-Integration on chains:Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients typically integers.The set of all k-chains forms a group...

      . In measure theory, by contrast, one interprets the integrand as a function f with respect to a measure and integrates over a subset A, without any notion of orientation; one writes to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher dimensional manifolds; see Differential form: Relation with measures for details.

      Fundamental theorem of calculus


      The fundamental theorem of calculus is the statement that differentiation
      Derivative
      In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

       and integration are inverse operations: if a continuous function
      Continuous function
      In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

       is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative
      Antiderivative
      In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

       of the function to be integrated.

      Statements of theorems

      • Fundamental theorem of calculus. Let ƒ be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by

      Then, F is continuous on [a, b], differentiable on the open interval (ab), and


      for all x in (a, b).
      • Second fundamental theorem of calculus. Let ƒ be a real-valued function defined on a closed interval [a, b] that admits an antiderivative
        Antiderivative
        In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...

         g on [ab]. That is, ƒ and g are functions such that for all x in [ab],



      If ƒ is integrable on [ab] then

      Improper integrals




      A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied. In some cases such integrals may be defined by considering the limit
      Limit (mathematics)
      In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

       of a sequence
      Sequence
      In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

       of proper Riemann integral
      Riemann integral
      In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...

      s on progressively larger intervals.

      If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity.
      If the integrand is only defined or finite on a half-open interval, for instance (a,b], then again a limit may provide a finite result.

      That is, the improper integral is the limit
      Limit (mathematics)
      In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

       of proper integrals as one endpoint of the interval of integration approaches either a specified real number
      Real number
      In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

      , or ∞, or −∞. In more complicated cases, limits are required at both endpoints, or at interior points.

      Consider, for example, the function integrated from 0 to ∞ (shown right). At the lower bound, as x goes to 0 the function goes to ∞, and the upper bound is itself ∞, though the function goes to 0. Thus this is a doubly improper integral. Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of . To integrate from 1 to ∞, a Riemann sum is not possible. However, any finite upper bound, say t (with t > 1), gives a well-defined result, . This has a finite limit as t goes to infinity, namely . Similarly, the integral from 13 to 1 allows a Riemann sum as well, coincidentally again producing . Replacing 13 by an arbitrary positive value s (with s < 1) is equally safe, giving . This, too, has a finite limit as s goes to zero, namely . Combining the limits of the two fragments, the result of this improper integral is
      This process does not guarantee success; a limit may fail to exist, or may be unbounded. For example, over the bounded interval 0 to 1 the integral of does not converge; and over the unbounded interval 1 to ∞ the integral of does not converge.
      It may also happen that an integrand is unbounded at an interior point, in which case the integral must be split at that point, and the limit integrals on both sides must exist and must be bounded. Thus
      But the similar integral
      cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value
      Cauchy principal value
      In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...

      .)

      Multiple integration




      Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written:


      Here x need not be a real number, but can be another suitable quantity, for instance, a vector in R3. Fubini's theorem
      Fubini's theorem
      In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...

       shows that such integrals can be rewritten as an iterated integral
      Multiple integral
      The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...

      . In other words, the integral can be calculated by integrating one coordinate at a time.

      Just as the definite integral of a positive function of one variable represents the area
      Area
      Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

       of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume
      Volume
      Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

       of the region between the surface defined by the function and the plane which contains its domain
      Domain (mathematics)
      In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

      . (The same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above mentioned region between the surface and the plane.) If the number of variables is higher, then the integral represents a hypervolume, a volume of a solid of more than three dimensions that cannot be graphed.

      For example, the volume of the cuboid
      Cuboid
      In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

       of sides 4 × 6 × 5 may be obtained in two ways:
      • By the double integral
      of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the cuboid. For example, if a rectangular base of such a cuboid is given via the xy inequalities 3 ≤ x ≤ 7, 4 ≤ y ≤ 10, our above double integral now reads


      From here, integration is conducted with respect to either x or y first; in this example, integration is first done with respect to x as the interval corresponding to x is the inner integral. Once the first integration is completed via the method or otherwise, the result is again integrated with respect to the other variable. The result will equate to the volume under the surface.

      • By the triple integral
      of the constant function 1 calculated on the cuboid itself.

      Line integrals




      The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector field
      Vector field
      In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

      s.

      A line integral (sometimes called a path integral) is an integral where the function
      Function (mathematics)
      In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

       to be integrated is evaluated along a curve
      Curve
      In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

      . Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.

      The function to be integrated may be a scalar field
      Scalar field
      In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

       or a vector field
      Vector field
      In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

      . The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length
      Arc length
      Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...

       or, for a vector field, the scalar product
      Inner product space
      In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

       of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on interval
      Interval (mathematics)
      In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

      s. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work
      Mechanical work
      In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

       is equal to force
      Force
      In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

      , F, multiplied by displacement, s, may be expressed (in terms of vector quantities) as:
      For an object moving along a path in a vector field
      Vector field
      In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

        such as an electric field
      Electric field
      In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

       or gravitational field
      Gravitational field
      The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

      , the total work done by the field on the object is obtained by summing up the differential work done in moving from to . This gives the line integral

      Surface integrals




      A surface integral is a definite integral taken over a surface
      Surface
      In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

       (which may be a curved set in space
      Space
      Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

      ); it can be thought of as the double integral
      Multiple integral
      The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...

       analog of the line integral
      Line integral
      In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

      . The function to be integrated may be a scalar field
      Scalar field
      In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

       or a vector field
      Vector field
      In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

      . The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.

      For an example of applications of surface integrals, consider a vector field v on a surface S; that is, for each point x in S, v(x) is a vector. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux
      Flux
      In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

       is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product
      Dot product
      In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

       of v with the unit surface normal
      Surface normal
      A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

       to S at each point, which will give us a scalar field, which we integrate over the surface:
      The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics
      Physics
      Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

      , particularly with the classical theory of electromagnetism
      Electromagnetism
      Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

      .

      Integrals of differential forms


      A differential form
      Differential form
      In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

       is a mathematical concept in the fields of multivariable calculus
      Multivariable calculus
      Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....

      , differential topology
      Differential topology
      In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

       and tensor
      Tensor
      Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

      s. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products
      Exterior algebra
      In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

       of exterior derivative
      Exterior derivative
      In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

      s forming an exterior algebra
      Exterior algebra
      In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

      , was introduced by Élie Cartan
      Élie Cartan
      Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

      .

      We initially work in an open set
      Open set
      The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

       in Rn.
      A 0-form is defined to be a smooth function
      Smooth function
      In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

       f.
      When we integrate a function
      Function (mathematics)
      In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

       f over an m-dimension
      Dimension
      In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

      al subspace S of Rn, we write it as

      (The superscripts are indices, not exponents.) We can consider dx1 through dxn to be formal objects themselves, rather than tags appended to make integrals look like Riemann sum
      Riemann sum
      In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....

      s. Alternatively, we can view them as covectors
      One-form
      In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...

      , and thus a measure
      Measure (mathematics)
      In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

       of "density" (hence integrable in a general sense). We call the dx1, …,dxn basic 1-forms
      One-form
      In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...

      .

      We define the wedge product
      Exterior algebra
      In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...

      , "∧", a bilinear "multiplication" operator on these elements, with the alternating property that


      for all indices a. Note that alternation along with linearity and associativity implies dxbdxa = −dxadxb. This also ensures that the result of the wedge product has an orientation
      Orientation (mathematics)
      In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

      .

      We define the set of all these products to be basic 2-forms, and similarly we define the set of products of the form dxadxbdxc to be basic 3-forms. A general k-form is then a weighted sum of basic k-forms, where the weights are the smooth functions f. Together these form a vector space
      Vector space
      A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

       with basic k-forms as the basis vectors, and 0-forms (smooth functions) as the field of scalars. The wedge product then extends to k-forms in the natural way. Over Rn at most n covectors can be linearly independent, thus a k-form with k > n will always be zero, by the alternating property.

      In addition to the wedge product, there is also the exterior derivative
      Exterior derivative
      In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

       operator d. This operator maps k-forms to (k+1)-forms. For a k-form ω = f dxa over Rn, we define the action of d by:


      with extension to general k-forms occurring linearly.

      This more general approach allows for a more natural coordinate-free approach to integration on manifold
      Manifold
      In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

      s. It also allows for a natural generalisation of the fundamental theorem of calculus
      Fundamental theorem of calculus
      The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

      , called Stokes' theorem
      Stokes' theorem
      In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

      , which we may state as


      where ω is a general k-form, and ∂Ω denotes the boundary
      Boundary (topology)
      In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

       of the region Ω. Thus, in the case that ω is a 0-form and Ω is a closed interval of the real line, this reduces to the fundamental theorem of calculus
      Fundamental theorem of calculus
      The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

      . In the case that ω is a 1-form and Ω is a two-dimensional region in the plane, the theorem reduces to Green's theorem
      Green's theorem
      In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...

      . Similarly, using 2-forms, and 3-forms and Hodge dual
      Hodge dual
      In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

      ity, we can arrive at Stokes' theorem
      Stokes' theorem
      In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

       and the divergence theorem
      Divergence theorem
      In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

      . In this way we can see that differential forms provide a powerful unifying view of integration.

      Summations


      The discrete equivalent of integration is summation
      Summation
      Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...

      . Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time scale calculus
      Time scale calculus
      In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid discrete–continuous dynamical systems...

      .

      Computing integrals


      The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus
      Fundamental theorem of calculus
      The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

      . Let f(x) be the function of x to be integrated over a given interval [a, b]. Then, find an antiderivative of f; that is, a function F such that F' = f on the interval. Provided the integrand and integral have no singularities
      Mathematical singularity
      In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

       on the path of integration, by the fundamental theorem of calculus,

      The integral is not actually the antiderivative, but the fundamental theorem provides a way to use antiderivatives to evaluate definite integrals.

      The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable. Techniques include:
      • Integration by substitution
        Integration by substitution
        In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...

      • Integration by parts
        Integration by parts
        In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

      • Changing the order of integration
        Order of integration (calculus)
        In calculus, interchange of the order of integration is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed...

      • Integration by trigonometric substitution
        Trigonometric substitution
        In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions:...

      • Integration by partial fractions
        Partial fractions in integration
        In integral calculus, partial fraction expansions provide an approach to integrating a general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions...

      • Integration by reduction formulae
        Integration by reduction formulae
        Integration by reduction formulae can be used when we want to integrate a function raised to the power n. If we have such an integral we can establish a reduction formula which can be used to calculate the integral for any value of n....

      • Integration using parametric derivatives
        Integration using parametric derivatives
        In mathematics, integration by parametric derivatives is a method of integrating certain functions.For example, suppose we want to find the integral...

      • Integration using Euler's formula
      • Differentiation under the integral sign
      • Contour integration
        Methods of contour integration
        In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....


      Alternate methods exist to compute more complex integrals. Many nonelementary integral
      Nonelementary integral
      In mathematics, a nonelementary antiderivative is an antiderivative for which it can be shown that there exists no formula in terms of elementary functions . A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist...

      s can be expanded in a Taylor series
      Taylor series
      In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

       and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-function
      Meijer G-Function
      In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's...

      s can also be used, assuming that the integrand can be written as a product of Meijer G-functions. There are also many less common ways of calculating definite integrals; for instance, Parseval's identity
      Parseval's identity
      In mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is thePythagorean theorem for inner-product spaces....

       can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral
      Gaussian integral
      The Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...

      .

      Computations of volumes of solids of revolution
      Solid of revolution
      In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line that lies on the same plane....

       can usually be done with disk integration
      Disk integration
      Disk integration, , is a means of calculating the volume of a solid of revolution of a solid-state material, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks of infinitesimal thickness...

       or shell integration
      Shell integration
      Shell integration is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.It makes use of the so-called "representative cylinder"...

      .

      Specific results which have been worked out by various techniques are collected in the list of integrals.

      Symbolic algorithms


      Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computer
      Computer
      A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

      s, many professionals, educators, and students have turned to computer algebra system
      Computer algebra system
      A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...

      s that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration presents a special challenge in the development of such systems.

      A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. For instance, it is known that the antiderivatives of the functions exp ( x2), xx and sin x /x cannot be expressed in the closed form involving only rational
      Rational function
      In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

       and exponential
      Exponential function
      In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

       functions, logarithm
      Logarithm
      The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

      , trigonometric
      Trigonometric function
      In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

       and inverse trigonometric function
      Inverse trigonometric function
      In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...

      s, and the operations of multiplication and composition; in other words, none of the three given functions is integrable in elementary functions. Differential Galois theory
      Differential Galois theory
      In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...

       provides general criteria that allow one to determine whether the antiderivative of an elementary function is elementary. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may be still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists. The Risch algorithm
      Risch algorithm
      The Risch algorithm, named after Robert Henry Risch, is an algorithm for the calculus operation of indefinite integration . The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational...

      , implemented in Mathematica
      Mathematica
      Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

       and other computer algebra system
      Computer algebra system
      A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...

      s, does just that for functions and antiderivatives built from rational functions, radicals
      Nth root
      In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

      , logarithm, and exponential functions.

      Some special integrands occur often enough to warrant special study. In particular, it may be useful to have, in the set of antiderivatives, the special functions
      Special functions
      Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....

       of physics
      Physics
      Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

       (like the Legendre functions, the hypergeometric function, the Gamma function
      Gamma function
      In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

      , the Incomplete Gamma function
      Incomplete gamma function
      In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of...

       and so on - see Symbolic integration
      Symbolic integration
      In calculus symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f, i.e...

       for more details). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging.

      Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. Very complex formulae are unlikely to have closed-form antiderivatives, so how much of an advantage this presents is a philosophical question that is open for debate.

      Numerical quadrature


      The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating. Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow. This motivates the study and application of numerical methods for approximating integrals, which today use floating-point arithmetic
      Floating point
      In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

       on digital electronic computers. Many of the ideas arose much earlier, for hand calculations; but the speed of general-purpose computers like the ENIAC
      ENIAC
      ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

       created a need for improvements.

      The goals of numerical integration are accuracy, reliability, efficiency, and generality. Sophisticated methods can vastly outperform a naive method by all four measures . Consider, for example, the integral
      which has the exact answer 9425 = 3.76. (In ordinary practice the answer is not known in advance, so an important task — not explored here — is to decide when an approximation is good enough.) A “calculus book” approach divides the integration range into, say, 16 equal pieces, and computes function values.
      Spaced function values
      x −2.00 −1.50 −1.00 −0.50  0.00  0.50  1.00  1.50  2.00
      f(x)  2.22800  2.45663  2.67200  2.32475  0.64400 −0.92575 −0.94000 −0.16963  0.83600
      x   −1.75 −1.25 −0.75 −0.25  0.25  0.75  1.25  1.75
      f(x)  2.33041  2.58562  2.62934  1.64019 −0.32444 −1.09159 −0.60387  0.31734


      Using the left end of each piece, the rectangle method
      Rectangle method
      In mathematics, specifically in integral calculus, the rectangle method computes an approximation to a definite integral, made by finding the area of a collection of rectangles whose heights are determined by the values of the function.Specifically, the interval over which the function is to be...

       sums 16 function values and multiplies by the step width, h, here 0.25, to get an approximate value of 3.94325 for the integral. The accuracy is not impressive, but calculus formally uses pieces of infinitesimal width, so initially this may seem little cause for concern. Indeed, repeatedly doubling the number of steps eventually produces an approximation of 3.76001. However, 218 pieces are required, a great computational expense for such little accuracy; and a reach for greater accuracy can force steps so small that arithmetic precision becomes an obstacle.

      A better approach replaces the horizontal tops of the rectangles with slanted tops touching the function at the ends of each piece. This trapezium rule is almost as easy to calculate; it sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width. This immediately improves the approximation to 3.76925, which is noticeably more accurate. Furthermore, only 210 pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy.

      Romberg's method
      Romberg's method
      In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule . The estimates generate a triangular array...

       builds on the trapezoid method to great effect. First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h0), T(h1), and so on, where hk+1 is half of hk. For each new step size, only half the new function values need to be computed; the others carry over from the previous size (as shown in the table above). But the really powerful idea is to interpolate
      Interpolation
      In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

       a polynomial through the approximations, and extrapolate to T(0). With this method a numerically exact answer here requires only four pieces (five function values)! The Lagrange polynomial
      Lagrange polynomial
      In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...

       interpolating {hk,T(hk)}k=0…2 = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76+0.148h2, producing the extrapolated value 3.76 at h = 0.

      Gaussian quadrature
      Gaussian quadrature
      In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration....

       often requires noticeably less work for superior accuracy. In this example, it can compute the function values at just two x positions, ±2√3, then double each value and sum to get the numerically exact answer. The explanation for this dramatic success lies in error analysis, and a little luck. An n-point Gaussian method is exact for polynomials of degree up to 2n−1. The function in this example is a degree 3 polynomial, plus a term that cancels because the chosen endpoints are symmetric around zero. (Cancellation also benefits the Romberg method.)

      Shifting the range left a little, so the integral is from −2.25 to 1.75, removes the symmetry. Nevertheless, the trapezoid method is rather slow, the polynomial interpolation method of Romberg is acceptable, and the Gaussian method requires the least work — if the number of points is known in advance. As well, rational interpolation can use the same trapezoid evaluations as the Romberg method to greater effect.
      Quadrature method cost comparison
      Method Trapezoid Romberg Rational Gauss
      Points 1048577 257 129 36
      Rel. Err. −5.3×10−13 −6.3×10−15 8.8×10−15 3.1×10−15
      Value


      In practice, each method must use extra evaluations to ensure an error bound on an unknown function; this tends to offset some of the advantage of the pure Gaussian method, and motivates the popular Gauss–Kronrod quadrature formulae. Symmetry can still be exploited by splitting this integral into two ranges, from −2.25 to −1.75 (no symmetry), and from −1.75 to 1.75 (symmetry). More broadly, adaptive quadrature
      Adaptive quadrature
      In applied mathematics, adaptive quadrature is a process in which the integral of a function f is approximated using static quadrature rules on adaptively refined subintervals of the integration domain...

       partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most.

      Simpson's rule
      Simpson's rule
      In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:...

      , named for Thomas Simpson
      Thomas Simpson
      Thomas Simpson FRS was a British mathematician, inventor and eponym of Simpson's rule to approximate definite integrals...

       (1710–1761), uses a parabolic curve to approximate integrals. In many cases, it is more accurate than the trapezoidal rule and others. The rule states that
      with an error of

      The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as Monte Carlo integration.

      A calculus text is no substitute for numerical analysis, but the reverse is also true. Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals. For example, improper integrals may require a change of variable or methods that can avoid infinite function values, and known properties like symmetry and periodicity may provide critical leverage.

      Practical applications



      Area under the curve (abbreviated AUC) is frequently used in pharmacokinetics
      Pharmacokinetics
      Pharmacokinetics, sometimes abbreviated as PK, is a branch of pharmacology dedicated to the determination of the fate of substances administered externally to a living organism...

       for functions where the x-axis represents time and the y-axis represents drug
      Drug
      A drug, broadly speaking, is any substance that, when absorbed into the body of a living organism, alters normal bodily function. There is no single, precise definition, as there are different meanings in drug control law, government regulations, medicine, and colloquial usage.In pharmacology, a...

       concentration. For such functions, the area under the curve usually correlates fairly well with the total effect on the body that the drug will have. In standard use, AUC is defined as either:
      • AUC, the integral after a single dose with a hypothetical infinite x-axis
      • AUCτ, the integral in the time interval between doses given regularly, and after having reached steady state.

      External links



      Online books