Fundamental Theorem of Calculus

 

Area function

We have already defined, for a continuous function f(x) on a closed interval [a, b] as the area of the region bounded by the curve y = f(x), X-axis and x= a and x = b.

In other words, area of the shaded region is a function of x.

The function A(x) is shown in figure below.

This area function A(x) is the anti derivative of f(x). That is f(x) = A'(x)

We state fundamental theorems of integral calculus without proof as they are beyond syllabus.

First Fundamental Theorem of Integral Calculus

Let f(x) be a continuous function on the closed interval [a, b].

Let the area function A(x) be defined by

then

Second Fundamental Theorem of Integral Calculus

Let f(x) be a continuous function defined on an interval [a,b].

between the limits a and b. This statement is also known as 'fundamental theorem of calculus'.

We call b, the upper limit of x and a, the lower limit.

If in place of F(x) we take F(x)+c as the value of the integral, we have

= [F(b) + c] - [F(a) + c] = F(b) + c - F(a) - c = F(b) - F(a)

Thus, the value of a definite integral is unique. It does not depend on the constant c and so in the evaluation of a definite integral the constant of integration does not play any role.

Note:

From the above two theorem, we infer the following

(Anti derivative of the function f(x) at b)

- (Anti derivative of the function f(x) at a)

(ii) The fundamental theorem of integral calculus shows a close relationship between differentiation and integration

(iii) These theorems give an alternate method evaluating definite integral, without calculating the limit of a sum.

Example:

Evaluate the definite integral of the following

Solution: