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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.


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)
(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:




