Fundamental theorem of calculus

Last updated

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.


The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also known as an indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions. [1]

Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration avoids numerical integration to compute integrals.


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 velocities) are actually closely related.

The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, [2] was by James Gregory (1638–1675). [3] [4] Isaac Barrow (1630–1677) proved a more generalized version of the theorem, [5] 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 area shaded in red stripes is close to h times f(x). Alternatively, if the function A(x) were known, this area would be exactly A(x + h) - A(x). These two values are approximately equal, particularly for small h. FTC geometric.svg
The area shaded in red stripes is close to h times f(x). Alternatively, if the function A(x) were known, this area would be exactly A(x + h) − A(x). These two values are approximately equal, particularly for small h.

For a continuous function y = f(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x. The function A(x) may not be known, but it is given that it represents the area under the curve.

The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this “strip” would be A(x + h) − A(x).

There is another way to estimate the area of this same strip. As shown in the accompanying figure, h is multiplied by f(x) 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 portion of the "excess" area shown in the diagram. So:

Rearranging terms:


As h approaches 0 in the limit, the last fraction can be shown to go to zero. [6] This is true because the area of the red portion of excess region is less than or equal to the area of the tiny black-bordered rectangle. More precisely,

where and are points where f reaches its maximum and its minimum, respectively, in the interval [x, x + h]. By the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does, which implies

This implies f(x) = A′(x). That is, the derivative of the area function A(x) exists and is the original function f(x); so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and finding the area under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus.

Physical intuition

Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other variable) adds up to the net change in the quantity.

Imagine for example using a stopwatch to mark-off tiny increments of time as a car travels down a highway. Imagine also looking at the car's speedometer as it travels, so that at every moment you know the velocity of the car. To understand the power of this theorem, imagine also that you are not allowed to look out of the window of the car, so that you have no direct evidence of how far the car has traveled.

For any tiny interval of time in the car, you could calculate how far the car has traveled in that interval by multiplying the current speed of the car times the length of that tiny interval of time. (This is because distance = speedtime.)

Now imagine doing this instant after instant, so that for every tiny interval of time you know how far the car has traveled. In principle, you could then calculate the total distance traveled in the car (even though you've never looked out of the window) by simply summing-up all those tiny distances.

distance traveled = the velocity at any instant a tiny interval of time

In other words,

distance traveled =

On the right hand side of this equation, as becomes infinitesimally small, the operation of "summing up" corresponds to integration. So what we've shown is that the integral of the velocity function can be used to compute how far the car has traveled.

Now remember that the velocity function is simply the derivative of the position function. So what we have really shown is that integrating the velocity simply recovers the original position function. This is the basic idea of the theorem: that integration and differentiation are closely related operations, each essentially being the inverse of the other.

In other words, in terms of one's physical intuition, the theorem simply states that the sum of the changes in a quantity over time (such as position, as calculated by multiplying velocity times time) adds up to the total net change in the quantity. Or to put this more generally:

then the idea that "distance equals speed times time" corresponds to the statement

meaning that one can recover the original function by integrating its derivative, the velocity , over .

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. [7]

Let f 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 uniformly continuous on [a, b] and differentiable on the open interval (a, b), and

for all x in (a, b).


Fundamental theorem of calculus (animation) Fundamental theorem of calculus (animation ).gif
Fundamental theorem of calculus (animation)

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 [8] or the Newton–Leibniz axiom.

Let be a real-valued function on a closed interval and 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 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 f(t), define the function F(x) as

For any two numbers x1 and x1 + Δx in [a, b], we have


Subtracting the two equalities gives

It can be shown that

(The sum of the areas of two adjacent regions is equal to the area of both regions combined.)

Manipulating this equation gives

Substituting the above into (1) results in

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

To keep the notation simple, we write just , but one should keep in mind that, for a given function , the value of depends on and on but is always confined to the interval . Substituting the above into (2) we get

Dividing both sides by gives

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

Take the limit as → 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 use the squeeze theorem. The number c is in the interval [x1, x1+ Δx], so x1cx1+ Δx.

Also, and

Therefore, according to the squeeze theorem,

The function f is continuous at x1, the limit can be taken inside the function:

Substituting into (3), we get

which completes the proof. [9] [ page needed ]

Proof of the corollary

Suppose F is an antiderivative of f, with f continuous on [a, b]. Let


By the first part of the theorem, we know G is also an antiderivative of f. Since F′ − G′ = 0 the mean value theorem implies that FG is a constant function, that is, there is a number c such that G(x) = F(x)+c for all x in [a, b]. Letting x=a, we have

which means c = −F(a). In other words, G(x) = F(x) − F(a), and so

Proof of the second part

This is a limit proof by Riemann sums. Let f be (Riemann) integrable on the interval [a, b], and let f admit an antiderivative F on [a, b]. Begin with the quantity F(b) − F(a). Let there be numbers x1, ..., xn such that

It follows that

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

The above quantity can be written as the following sum:

Next, we employ the mean value theorem. Stated briefly,

Let F be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there exists some c in (a, b) 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 [xi−1, xi]. According to the mean value theorem (above),

Substituting the above into (1), we get

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

A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function. Riemann integral irregular.gif
A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function.

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 i, or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with n rectangles. Now, as the size of the partitions get smaller and n 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 f 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 (2). This gives us

Neither F(b) nor F(a) is dependent on , so the limit on the left side remains F(b) − F(a).

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

which completes the proof.

It almost looks like the first part of the theorem follows directly from the second. That is, suppose G is an antiderivative of f. Then by the second theorem, . Now, suppose . Then F has the same derivative as G, and therefore F′ = f. This argument only works, however, if we already know that f has an antiderivative, and the only way we know that all continuous functions have antiderivatives is by the first part of the Fundamental Theorem. [1] For example, if f(x) = ex2, then f 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).


As an example, suppose the following is to be calculated:

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

Or, more generally, suppose that

is to be calculated. Here, and can be used as the antiderivative. Therefore:

Or, equivalently,

As a theoretical example, the theorem can be used to prove that


the result follows from,


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 function on [a, b] and x0 is a number in [a, b] such that f is continuous at x0, then

is differentiable for x = x0 with F′(x0) = f(x0). 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′(x) = f(x) 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. [10]

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

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


This result may fail for continuous functions F that admit a derivative f(x) at almost every point x, as the example of the Cantor function shows. However, if F is absolutely continuous, it admits a derivative F′(x) at almost every point x, and moreover F′ is integrable, with F(b) − F(a) equal to the integral of F′ on [a, b]. Conversely, if f is any integrable function, then F as given in the first formula will be absolutely continuous with F′ = f a.e.

The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function F(x) admits a derivative f(x) at all but countably many points, then f(x) is Henstock–Kurzweil integrable and F(b) − F(a) is equal to the integral of f on [a, b]. The difference here is that the integrability of f does not need to be assumed. [12]

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 U is an open set in C and f : UC is a function that has a holomorphic antiderivative F on U. Then for every curve γ : [a, b] → U, 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): [13] Let M be an oriented piecewise smooth manifold of dimension n and let be a smooth compactly supported (n1)-form on M. If ∂M denotes the boundary of M given its induced orientation, then

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

The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold (e.g. Rk) 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


    Related Research Articles

    Antiderivative Concept in calculus

    In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation, and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.

    Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

    Integral Operation in calculus

    In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

    In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable . The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication.

    Riemann integral Basic Integral in Elementary Calculus

    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. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

    Dirac delta function Pseudo-function δ such that an integral of δ(x-c)f(x) always takes the value of f(c)

    In mathematics, the Dirac delta function is a generalized function or distribution introduced by physicist Paul Dirac. It is called a function, although it is not a function on the level one would expect, that is, it is not a function RC, but a function on the space of test functions. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

    Differential calculus Area of mathematics; subarea of calculus

    In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

    In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.

    In calculus, the constant of integration, often denoted by , is a constant added to the end of an antiderivative of a function to indicate that the indefinite integral of , on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.

    Riemann sum Approximation technique in integral calculus

    In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.

    In calculus, integration by substitution, also known as u-substitution or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards".

    In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.

    In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral, Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.

    In mathematics, the Riemann–Liouville integral associates with a real function another function Iα f of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of α, Iα f is an iterated antiderivative of f of order α. The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions. It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential.

    In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

    Inexact differential

    An inexact differential or imperfect differential is a type of differential used in thermodynamics to express changes in path dependent quantities. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential; i.e. its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions.

    In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.

    In many applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions of a particular parameter. In physical applications, that parameter is frequently time t.

    In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse of a continuous and invertible function , in terms of and an antiderivative of . This formula was published in 1905 by Charles-Ange Laisant.

    Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.


    1. 1 2 Spivak, Michael (1980), Calculus (2nd ed.), Houston, Texas: Publish or Perish Inc.
    2. Malet, Antoni (1993). "James Gregorie on tangents and the "Taylor" rule for series expansions". Archive for History of Exact Sciences . Springer-Verlag. doi:10.1007/BF00375656. Gregorie's thought, on the other hand, belongs to a conceptual framework strongly geometrical in character. (page 137)
    3. See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, Sherlock Holmes in Babylon and Other Tales of Mathematical History, Mathematical Association of America, 2004, p. 114.
    4. Gregory, James (1668). Geometriae Pars Universalis. Museo Galileo: Patavii: typis heredum Pauli Frambotti.
    5. Child, James Mark; Barrow, Isaac (1916). The Geometrical Lectures of Isaac Barrow. Chicago: Open Court Publishing Company.
    6. Bers, Lipman. Calculus, pp. 180–181 (Holt, Rinehart and Winston (1976).
    7. Apostol 1967 , §5.1
    8. Apostol 1967 , §5.3
    9. Leithold, 1996.
    10. Bartle (2001), Thm. 4.11.
    11. Rudin 1987 , th. 7.21
    12. Bartle (2001), Thm. 4.7.
    13. Spivak, M. (1965). Calculus on Manifolds. New York: W. A. Benjamin. pp. 124–125. ISBN   978-0-8053-9021-6.


    Further reading