Fundamental theorem of calculus

The fundamental theorem of calculus is the statement that the two central operations of calculus Calculus

Calculus is a central branch of mathematics [i], developed from algebra [i] and geometry [i]. ...

, differentiation Derivative

In mathematics [i], the derivative is defined as the instantaneous rate of change of a function [i] ...

and integration Integral

In calculus [i], the integral of a function [i] is an extension of the concept of a sum. ...

, are inverse functions of one another. It is of such central importance in calculus that it is called the fundamental theorem for the entire field of study. This means that if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative Antiderivative

In calculus [i], an antiderivative, primitive or indefinite integral of a function [i] ...

of the function to be integrated.

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Encyclopedia

The fundamental theorem of calculus is the statement that the two central operations of calculus Calculus

Calculus is a central branch of mathematics [i], developed from algebra [i] and geometry [i]. ...

, differentiation Derivative

In mathematics [i], the derivative is defined as the instantaneous rate of change of a function [i] ...

and integration Integral

In calculus [i], an antiderivative, primitive or indefinite integral of a function [i] ...

of the function to be integrated. In his 2003 book , James Stewart credits the idea that led to the fundamental theorem to the English mathematician Isaac Barrow Isaac Barrow

Isaac Barrow was an English [i] divine, scholar and mathematician [i] who is generally given min ...

although the first known proof of the fundamental theorem was due to the Scottish mathematician James Gregory.

Intuition

Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time add up to the net change in the quantity.

To comprehend this statement, we will start with an example. Suppose a particle travels in a straight line with its position given by x where t is time. The derivative of this function is equal to the infinitesimal change in quantity per infinitesimal change in time . Let us define this change in distance per time as the speed v of the particle. In Leibniz's notation:

Rearranging that equation, it is clear that:

By the logic above, a change in x, call it , is the sum of the infinitesimal changes dx. It is also equal to the sum of the infinitesimal products of the derivative and time. This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative. As one can reasonably infer, this operation works in reverse as we can differentiate the result of our integral to recover the original function.

Formal statements

Stated formally, the theorem says the following.

Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined for x in [a, b] by
then
for every x in [a, b].

Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be a function such that
for all x in [a, b]
then
.

Corollary

Let f be a real-valued function defined on a closed interval [a, b]. Let F be a function such that
for all x in [a, b]
then
and
.

Examples

As an example, suppose you need to calculate

Here, and we can use as the antiderivative. Therefore:

Or, more generally, suppose you need to calculate

Here, and we can use as the antiderivative. Therefore:

But this result would have been found much more easily as

Proof

It is given that

Let there be two numbers x₁ and x₁ + ?x in [a, b]. So we have
and
.

Subtracting the two equations gives
.

It can be shown that
.

Manipulating this equation gives
.

Substituting the above into results in
.

According to the mean value theorem Mean value theorem

In calculus [i], the mean value theorem states, roughly, that given a section of a smooth curve, there i ...

for integration, there exists a c in [x₁, x₁ + ?x] such that
.

Substituting the above into we get
.

Dividing both sides by ?x gives
.

Notice that the expression on the left side of the equation is Newton's difference quotient for F at x₁.

Take the limit as ?x ? 0 on both sides of the equation.

The expression on the left side of the equation is the definition of the derivative of F at x₁.
.

To find the other limit, we will use the squeeze theorem Squeeze theorem

In calculus [i], the squeeze theorem is a theorem [i] regarding the limit of a function [i]. ...

. The number c is in the interval [x₁, x₁ + ?x], so x₁ = c = x₁ + ?x.

Also, and .

Therefore, according to the squeeze theorem,
.

Substituting into , we get
.

The function f is continuous at c, so the limit can be taken inside the function. Therefore, we get
.
which completes the proof.

Alternative proof

This is a limit proof by Riemann sums Riemann integral

In the branch of mathematics [i] known as real analysis [i], the Riemann integral, created by Bernhard Riemann [i]...

.

Let f be continuous on the interval [a, b], and let F be an antiderivative of f. Begin with the quantity
.

Let there be numbers

x₁, ..., x_n

such that

.

It follows that
.

Now, we add each F along with its additive inverse, so that the resulting quantity is equal:

The above quantity can be written as the following sum:

Next we will employ the mean value theorem Mean value theorem

In calculus [i], the mean value theorem states, roughly, that given a section of a smooth curve, there i ...

. Stated briefly,

Let F be continuous on the closed interval [a, b] and differentiable on the open interval . Then there exists some c in such that

It follows that

The function F is differentiable on the interval [a, b]; therefore, it is also differentiable and continuous on each interval x_i-1. Therefore, according to the mean value theorem ,

Substituting the above into , we get

The assumption implies Also, can be expressed as of partition .

Notice that we are describing the area of a rectangle, with the width times the height, and we are adding the areas together. Each rectangle, by virtue of the Mean Value Theorem Mean value theorem

In calculus [i], the mean value theorem states, roughly, that given a section of a smooth curve, there i ...

, describes an approximation of the curve section it is drawn over. Also notice that does not need to be the same for any value of , or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with rectangles. Now, as the size of the partitions get smaller and n increases, resulting in more partitions to cover the space, we will get closer and closer to the actual area of the curve.

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral Riemann integral

In the branch of mathematics [i] known as real analysis [i], the Riemann integral, created by Bernhard Riemann [i]...

. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.

So, we take the limit on both sides of . This gives us

Neither F nor F is dependent on ||?||, so the limit on the left side remains F - F.

The expression on the right side of the equation defines an integral over f from a to b. Therefore, we obtain
which completes the proof.

Generalizations

We don't need to assume continuity of f on the whole interval. Part I of the theorem then says: if f is any Lebesgue integrable Lebesgue integration

In mathematics [i], the integral [i] of a nonnegative function can be regarded in the simplest case as the ...

function on and is a number in such that is continuous at , then

is differentiable for with . We can relax the conditions on f still further and suppose that it is merely locally integrable. In that case, we can conclude that the function F is differentiable almost everywhere and F'=f almost everywhere. This is sometimes known as Lebesgue's differentiation theorem.

Part II of the theorem is true for any Lebesgue integrable function f which has an antiderivative F .

The version of Taylor's theorem Taylor's theorem

In calculus [i], Taylor's theorem, named after the mathematician [i] Brook Taylor [i], who stated it in ...

which expresses the error term as an integral can be seen as a generalization of the Fundamental Theorem.

There is a version of the theorem for complex Complex number

In mathematics [i], a complex number is a number [i] of the form
...

functions: suppose U is an open set in C and f: U -> C is a function which has a holomorphic antiderivative F on U. Then for every curve ? : [a, b] -> U, the curve integral Line integral

In mathematics [i], a line integral is an integral [i] where the function [i] to be integrated ...

can be computed as

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on manifold Manifold

A manifold is an abstract mathematical space [i] in which every point has a neighborho ...

s.

The most powerful statement in this direction is Stokes' theorem.

References

Larson, Ron, Bruce H. Edwards, David E. Heyd. Calculus of a single variable. 7th ed. Boston: Houghton Mifflin Company, 2002.
Leithold, L. . The calculus 7 of a single variable. 6th ed. New York: HarperCollins College Publishers.
Malet, A, Studies on James Gregorie .
Rosa, Don From Duckburg to Lillehammer, Donald Duck 283, Disney.
Stewart, J. . Fundamental Theorem of Calculus. In Integrals. In Calculus: early transcendentals. Belmont, California: Thomson/Brooks/Cole.

Turnbull, H W , The James Gregory Tercentenary Memorial Volume