The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... For a non-technical overview of the subject, see Calculus. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically...

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration^[1] can be reversed by a differentiation. In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals. This article deals with the concept of an integral in calculus. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638-1675)^[2]. Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716) independently developed the theorem in its final form. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... James Gregory For other people with the same name, see James Gregory. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1728) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... “Leibniz” redirects here. ...

Intuition

Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or some other quantity) add up to the net change in the quantity. Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

To comprehend this statement, we will start with an example. Suppose a particle travels in a straight line with its position given by x(t) where t is time. The derivative of this function is equal to the infinitesimal change in quantity per infinitesimal change in time (of course, the derivative itself is dependent on time). Let us define this change in distance per change in time as the speed v of the particle. In Leibniz's notation: In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Δx and Δy represent finite...

Rearranging this equation, it follows that: The differential dy In calculus, a differential is an infinitesimally small change in a variable. ...

By the logic above, a change in x, call it Δx, 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 derivative.

Formal statements

There are two parts to the Fundamental Theorem of Calculus. Loosely put, the first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivative and definite integral. In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... This article deals with the concept of an integral in calculus. ...

First part

This part is sometimes referred to as First Fundamental Theorem of Calculus.

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 In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

Then, for every x in [a, b],

.

The operation is a definite integral with variable upper limit, and its result F(x) is one of the infinitely many antiderivatives of f. This article deals with the concept of an integral in calculus. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

Second part

This part is sometimes referred to as Second Fundamental Theorem of Calculus.

Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be an antiderivative of f, that is one of the infinitely many functions such that, for all x in [a, b], In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

.

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], In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...

Then, for all x in [a, b],

and

.

Examples

As an example, suppose you need to calculate

Here, f(x) = x² and we can use as the antiderivative. Therefore:

Or, more generally, suppose you need to calculate

Here, f(t) = t³ and we can use as the antiderivative. Therefore:

But this result could have been found much more easily as

Proof

Suppose that

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

and

Subtracting the two equations 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 c in [x₁, x₁ + Δx] such that In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

.

Substituting the above into (2) we get

.

Dividing both sides by Δx 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 Δx → 0 on both sides of the equation. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

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. The number c is in the interval [x₁, x₁ + Δx], so x₁ ≤ c ≤ x₁ + Δx. In calculus, the squeeze theorem (also known as the pinching theorem or the sandwich theorem, sometimes the squeeze lemma) is a theorem regarding the limit of a function. ...

Also, and .

Therefore, according to the squeeze theorem,

Substituting into (3), 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.

(Leithold et al, 1996)

Alternative proof

This is a limit proof by Riemann sums. If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

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(x_i) 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. Stated briefly, In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

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 x_i-1. Therefore, according to the mean value theorem (above),

Substituting the above into (1), we get

The assumption implies F'(c_i) = f(c_i). Also, x_i − x_{i − 1} can be expressed as Δx of partition i.

A converging sequence of Riemann sums. The numbers in the upper right are the areas of the grey rectangles. They converge to the integral of the function.

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, describes an approximation of the curve section it is drawn over. Also notice that Δx_i does not need to be the same for any value 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 will get closer and closer to the actual area of the curve. Animation that illustrates features of Riemann integration File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Animation that illustrates features of Riemann integration File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral. 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. If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

So, we take the limit on both sides of (2). This gives us

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

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 function on [a,b] and x₀ is a number in [a,b] such that f is continuous at x₀, then The integral of a positive function can be interpreted as the area under a curve. ...

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. This is sometimes known as Lebesgue's differentiation theorem. In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

Part II of the theorem is true for any Lebesgue integrable function f which has an antiderivative F (not all integrable functions do, though).

The version of Taylor's theorem which expresses the error term as an integral can be seen as a generalization of the Fundamental Theorem. In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ...

There is a version of the theorem for complex 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 γ : [a, b] -> U, the curve integral can be computed as In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on manifolds. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...

One of the most powerful statements in this direction is Stokes' theorem. Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

See also

• Differintegral

In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. ...

Notes

1. ^ More exactly, the theorem deals with definite integration with variable upper limit and arbitrarily selected lower limit. This particular kind of definite integration allows us to compute one of the infinitely many antiderivatives of a function (except for those which do not have a zero). Hence, it is almost equivalent to indefinite integration, defined by most authors as an operation which yields any one of the possible antiderivatives of a function, including those without a zero.
2. ^ 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.

The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

References

• Larson, Ron, Bruce H. Edwards, David E. Heyd. Calculus of a single variable. 7th ed. Boston: Houghton Mifflin Company, 2002.
• Leithold, L. (1996). The calculus 7 of a single variable. 6th ed. New York: HarperCollins College Publishers.
• Malet, A, Studies on James Gregorie (1638-1675) (PhD Thesis, Princeton, 1989).
• Stewart, J. (2003). Fundamental Theorem of Calculus. In Integrals. In Calculus: early transcendentals. Belmont, California: Thomson/Brooks/Cole.

• Turnbull, H W (ed.), The James Gregory Tercentenary Memorial Volume (London, 1939)

External links

• James Gregory's Euclidean Proof of the Fundemental Theorem of Calculus at Convergence