Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.

The first part of the theorem, the first fundamental theorem of calculus, states that for a function  , an antiderivative or indefinite integral  may be obtained as the integral of  over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions.

Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function  over a fixed interval is equal to the change of any antiderivative  between the ends of the interval. This greatly simplifies the calculation of a definite integral provided an antiderivative can be found by symbolic integration, thus avoiding numerical integration.

History

The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration. The origins of differentiation likewise predate the fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of continuity of functions and motion were studied by the Oxford Calculators and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related.

From the conjecture and the proof of the fundamental theorem of calculus, calculus as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved a more generalized version of the theorem, while his student Isaac Newton (1642–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.

Geometric meaning

The first fundamental theorem may be interpreted as follows. Given a continuous function  whose graph is plotted as a curve, one defines a corresponding "area function"  such that  is the area beneath the curve between  and . The area  may be not easily computable, but it is assumed to be well defined.

The area under the curve between  and  could be computed by finding the area between  and , then subtracting the area between  and . In other words, the area of this "strip" would be .

There is another way to estimate the area of this same strip. As shown in the accompanying figure,  is multiplied by  to find the area of a rectangle that is approximately the same size as this strip. So:

In fact, this estimate becomes a perfect equality if we add the red "Excess" area in the diagram. So:

Rearranging terms:

As  approaches  in the limit, the last fraction must go to zero. To see this, note that the excess region is inside the tiny black-bordered rectangle, giving an upper bound for the excess area:where  and  are points where   reaches its maximum and its minimum, respectively, in the interval .

Thus:
By the continuity of , the right-hand expression tends to zero as  does. Therefore, the left-hand side also tends to zero, and:

That is, the derivative of the area function  exists and is equal to the original function , so the area function is an antiderivative of the original function. 

Thus, the derivative of the integral of a function (the area) is the original function, so that derivative and integral are inverse operations which reverse each other. This is the essence of the Fundamental Theorem.

Physical intuition
Intuitively, the fundamental theorem states that integration and differentiation are essentially inverse operations which reverse each other.

The second fundamental theorem says that the sum of infinitesimal changes in a quantity over time (the integral of the derivative of the quantity) adds up to the net change in the quantity. To visualize this, imagine traveling in a car and wanting to know the distance traveled (the net change in position along the highway). You can see the velocity on the speedometer but cannot look out to see your location. Each second, you can find how far the car has traveled using , multiplying the current speed (in kilometers or miles per hour) by the time interval (1 second =  hour). Summing up all these small steps, you can calculate the total distance traveled, without ever looking outside the car:As  becomes infinitesimally small, the summing up corresponds to integration. Thus, the integral of the velocity function (the derivative of position) computes how far the car has traveled (the net change in position).

The first fundamental theorem says that any quantity is the rate of change (the derivative) of the integral of the quantity from a fixed time up to a variable time. Continuing the above example, if you imagine a velocity function, you can integrate it from the starting time up to any given time to obtain a distance function whose derivative is the given velocity. (To obtain the highway-marker position, you need to add your starting position to this integral.)

Formal statements
There are two parts to the theorem. The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

First part
This part is sometimes referred to as the first fundamental theorem of calculus.

Let  be a continuous real-valued function defined on a closed interval . Let  be the function defined, for all  in , by

Then  is uniformly continuous on  and differentiable on the open interval , and

for all  in  so  is an antiderivative of .

Corollary

The fundamental theorem is often employed to compute the definite integral of a function  for which an antiderivative  is known. Specifically, if  is a real-valued continuous function on  and  is an antiderivative of  in  then

The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following part of the theorem.

Second part
This part is sometimes referred to as the second fundamental theorem of calculus or the Newton–Leibniz theorem.

Let  be a real-valued function on a closed interval  and  a continuous function on  which is an antiderivative of  in :

If  is Riemann integrable on  then

The second part is somewhat stronger than the corollary because it does not assume that  is continuous.

When an antiderivative  of  exists, then there are infinitely many antiderivatives for , obtained by adding an arbitrary constant to . Also, by the first part of the theorem, antiderivatives of  always exist when  is continuous.

Proof of the first part
For a given function , define the function  as

For any two numbers  and  in , we have

the latter equality resulting from the basic properties of integrals and the additivity of areas.

According to the mean value theorem for integration, there exists a real number  such that

It follows that

and thus that

Taking the limit as  and keeping in mind that  one gets 

that is, 

according to the definition of the derivative, the continuity of , and the squeeze theorem.

Proof of the corollary
Suppose  is an antiderivative of , with  continuous on . Let

By the first part of the theorem, we know  is also an antiderivative of . Since  the mean value theorem implies that  is a constant function, that is, there is a number  such that  for all  in . Letting , we have

which means . In other words, , and so

Proof of the second part
This is a limit proof by Riemann sums.

To begin, we recall the mean value theorem. Stated briefly, if  is continuous on the closed interval  and differentiable on the open interval , then there exists some  in  such that

Let  be (Riemann) integrable on the interval , and let  admit an antiderivative  on  such that  is continuous on . Begin with the quantity . Let there be numbers  such that

It follows that

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

The above quantity can be written as the following sum:

The function  is differentiable on the interval  and continuous on the closed interval ; therefore, it is also differentiable on each interval  and continuous on each interval . According to the mean value theorem (above), for each  there exists a  in  such that

Substituting the above into (), we get

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

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, describes an approximation of the curve section it is drawn over. Also  need not be the same for all values 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  increases, resulting in more partitions to cover the space, we 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. We know that this limit exists because  was assumed to be integrable. 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  nor  is dependent on , so the limit on the left side remains .

The expression on the right side of the equation defines the integral over  from  to . Therefore, we obtain

which completes the proof.

Relationship between the parts

As discussed above, a slightly weaker version of the second part follows from the first part.

Similarly, it almost looks like the first part of the theorem follows directly from the second. That is, suppose  is an antiderivative of . Then by the second theorem, . Now, suppose . Then  has the same derivative as , and therefore . This argument only works, however, if we already know that  has an antiderivative, and the only way we know that all continuous functions have antiderivatives is by the first part of the Fundamental Theorem.
For example, if , then  has an antiderivative, namely

and there is no simpler expression for this function. It is therefore important not to interpret the second part of the theorem as the definition of the integral. Indeed, there are many functions that are integrable but lack elementary antiderivatives, and discontinuous functions can be integrable but lack any antiderivatives at all. Conversely, many functions that have antiderivatives are not Riemann integrable (see Volterra's function).

Examples

Computing a particular integral
Suppose the following is to be calculated:

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

Using the first part
Suppose

is to be calculated. Using the first part of the theorem with  gives

This can also be checked using the second part of the theorem. Specifically,  is an antiderivative of , so

An integral where the corollary is insufficient
Suppose

Then  is not continuous at zero. Moreover, this is not just a matter of how  is defined at zero, since the limit as  of  does not exist. Therefore, the corollary cannot be used to compute

But consider the function

Notice that  is continuous on  (including at zero by the squeeze theorem), and  is differentiable on  with  Therefore, part two of the theorem applies, and

Theoretical example
The theorem can be used to prove that

Since, 

the result follows from,

Generalizations
The function  does not have to be continuous over the whole interval. Part I of the theorem then says: if  is any Lebesgue integrable function on  and  is a number in  such that  is continuous at , then

is differentiable for  with . We can relax the conditions on  still further and suppose that it is merely locally integrable. In that case, we can conclude that the function  is differentiable almost everywhere and  almost everywhere. On the real line this statement is equivalent to Lebesgue's differentiation theorem. These results remain true for the Henstock–Kurzweil integral, which allows a larger class of integrable functions.

In higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every , the average value of a function  over a ball of radius  centered at  tends to  as  tends to 0.

Part II of the theorem is true for any Lebesgue integrable function , which has an antiderivative  (not all integrable functions do, though). In other words, if a real function  on  admits a derivative  at every point  of  and if this derivative  is Lebesgue integrable on , then

This result may fail for continuous functions  that admit a derivative  at almost every point , as the example of the Cantor function shows. However, if  is absolutely continuous, it admits a derivative  at almost every point , and moreover  is integrable, with  equal to the integral of  on . Conversely, if  is any integrable function, then  as given in the first formula will be absolutely continuous with  almost everywhere.

The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function  admits a derivative  at all but countably many points, then  is Henstock–Kurzweil integrable and  is equal to the integral of  on . The difference here is that the integrability of  does not need to be assumed.

The version of Taylor's theorem, 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 functions: suppose  is an open set in  and  is a function that has a holomorphic antiderivative  on . Then for every curve , the curve integral can be computed as

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on manifolds. One such generalization offered by the calculus of moving surfaces is the time evolution of integrals. The most familiar extensions of the fundamental theorem of calculus in higher dimensions are the divergence theorem and the gradient theorem.

One of the most powerful generalizations in this direction is Stokes' theorem (sometimes known as the fundamental theorem of multivariable calculus): Let  be an oriented piecewise smooth manifold of dimension  and let  be a smooth compactly supported -form on . If  denotes the boundary of  given its induced orientation, then

Here  is the exterior derivative, which is defined using the manifold structure only.

The theorem is often used in situations where  is an embedded oriented submanifold of some bigger manifold (e.g. ) on which the form  is defined.

The fundamental theorem of calculus allows us to pose a definite integral as a first-order ordinary differential equation.

can be posed as

with  as the value of the integral.

See also

 Differentiation under the integral sign
 Telescoping series
 Fundamental theorem of calculus for line integrals
 Notation for differentiation

Notes

References

Bibliography

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Further reading

 .
 .
 Malet, A., Studies on James Gregorie (1638-1675) (PhD Thesis, Princeton, 1989).
 Hernandez Rodriguez, O. A.; Lopez Fernandez, J. M. . "Teaching the Fundamental Theorem of Calculus: A Historical Reflection", Loci: Convergence (MAA), January 2012.
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External links

 
 James Gregory's Euclidean Proof of the Fundamental Theorem of Calculus at Convergence
 Isaac Barrow's proof of the Fundamental Theorem of Calculus
 Fundamental Theorem of Calculus at imomath.com
 Alternative proof of the fundamental theorem of calculus
 Fundamental Theorem of Calculus MIT.
 Fundamental Theorem of Calculus Mathworld.

Articles containing proofs
Theorems in calculus
Theorems in real analysis