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The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. 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 of the function to be integrated.
Intuition
Intuitively, the theorem simply says that the sum of infinitesimal changes in a quantity over time (or some other quantity) add up to the net change in quantity.
To get a feeling for the 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 x per infinitesimal change in time (of course, the derivative itself is dependent on time). Let us define this change in distance per time as the speed v of the particle. In Leibniz's notation:
- <math>\frac{dx}{dt} = v(t) <math>
Rearranging that equation, it is clear that:
- <math>dx = v(t)\,dt <math>
By the logic above, a change in x, call it <math>\Delta x<math>, 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. Clearly, this operation works in reverse as we can differentiate the result of our integral to recover the speed function.
Formal statements
Stated formally, the theorem says:
If the function g(x) is continuous on some interval [a, b], then there exist infinitely many antiderivatives G(x) whose derivatives are g(x).
If the function f' (x) is continuous on some interval [p, q] and f(x) is one if its antiderivatives, then
- <math>\int_a^b f'(x) dx = f(b) - f(a) = \Delta f(x)<math>
- <math>f(x) = \int_a^x f'(t)\, dt + f(a)<math>
if a, b, and x are in [p, q].
Differentiating both sides, we find:
- <math>f'(x) = \frac{d}{dx} \int_a^x f'(t)\, dt.<math>
As an example, suppose you need to calculate
- <math>\int_2^5 x^2\, dx <math>
Here, <math>f(x) = x^2<math> and we can use <math>F(x) = (1/3) x^3<math> as antiderivative. Therefore:
- <math>\int_2^5 x^2\, dx = F(5) - F(2) = {125 \over 3} - {8 \over 3} = {117 \over 3} = 39.<math>
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 <math>[a, b]<math> and <math>x_0<math> is a number in <math>[a, b]<math> such that <math>f<math> is continuous at <math>x_0<math>, then
- <math>F(x) = \int_a^x f(t)\, dt<math>
is differentiable for <math>x = x_0<math> with <math>F'(x_0) = f(x_0)<math>. 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.
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.
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
- <math>\oint_{\gamma} f(z) \,dz = F(\gamma(b)) - F(\gamma(a)).<math>
The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on manifolds.
The most powerful statement in this direction is Stokes' theorem.
de:Fundamentalsatz der Analysis
es:Teorema fundamental del cálculo integral
fr:théorème fondamental du calcul différentiel et intégral
ja:微積分学の基本定理
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