Definite Integrals


   
 
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:
 
 
 
 
 
 
 
     
   
Get FREE Live Tutoring
Get FREE Live Tutoring
(No credit card required)