An overview of the fundamental theorem of calculus

The second fundamental problem addressed by calculus is the area of a region of the plane bounded by various curves. Many practical problems in various disciplines require the evaluation of areas for their solution, and the solution to the problem of areas necessarily involves the notion of limits. On the surface the problem of areas appears unrelated to the problem of tangents. However, the two problems are very closely related. One is the inverse of the other. Finding area is equivalent to finding the anti derivative or finding the integral. The relationship between derivatives and integrals is called the fundamental calculus theorem.

We shall now demonstrate the relationship between definite integral and indefinite integral. A consequence of this relationship is that we will be able to calculate definite integrals of functions whose anti derivatives we can find.
1^st fundamental theorem of calculus:
Suppose that a function f is continuous on an interval I that contains the point a. Let the function F be defined on I  by:

Then F is differentiable on I and F’(x) = f(x) there. Thus, F is an anti derivative of f on I:

2nd fundamental theorem of calculus:
If G(x) is any anti derivative of f(x) on l, so that G’(x) is equal to f(x) on l, then for any b in l we have,

Fundamental theorem of calculus proof for part I:

Using the definition of derivatives we can calculate

Second fundamental theorem of calculus proof:
Continuing from the proof above, if G’(x) = f(x), then F(x) = G(x) + C on l for some constant C. Hence,

Let x = a and obtain 0 = G(a) + C, so C = -G(a). Now let x= b, then we get,

For fundamental theorem of calculus example, anywhere else, we can the t with x or any other variable as the variable of integration on the left hand side.