Topics in Calculus

Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Lists of integrals

Vector Calculus

Vector | Vector field | Matrix | Partial Derivative | Gradient | Flux | Divergence | Divergence Theorem | Del | Curl | Stokes' Theorem

Tensor Calculus

Tensor | Tensor field | Tensor product | Exterior power | Exterior Derivative | Covariant derivative | Manifold

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:

Rearranging that equation, it is clear that:

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. 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

if a, b, and x are in [p, q]. Differentiating both sides, we find:

As an example, suppose you need to calculate

Here, f(x) = x² and we can use F(x) = (1 / 3)x³ as antiderivative. Therefore:

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

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.

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

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.