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 and x0 is a number in 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 (Bartle 2001, Thm. 4.11).
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 will tend 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 admits a derivative f(x) at every point x of and if this derivative f is Lebesgue integrable on, 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. But the result remains true if F is absolutely continuous: in that case, F admits a derivative f(x) at almost every point x and, as in the formula above, F(b) − F(a) is equal to the integral of f on .
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 . The difference here is that the integrability of f does not need to be assumed. (Bartle 2001, Thm. 4.7)
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 : → 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 two dimensions are Green's theorem and the two-dimensional case of the Gradient theorem.
One of the most powerful statements in this direction is Stokes' theorem: Let M be an oriented piecewise smooth manifold of dimension n and let be an n−1 form that is a compactly supported differential form on M of class C1. If ∂M denotes the boundary of M with 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 on which the form is defined.
Read more about this topic: Fundamental Theorem Of Calculus