Page 71 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes  In case of definite integral as limit of sum, the areas of inner rectangles are
h

h
hf ( ),hf (a h ),hf (a  2 ),hf (a  3 ), ,hf (a n  1 ).
a

h
 The area In case of definite integral as limit of sum, is close to the area of the region
bounded by the curve y = f(x), x-axis and the ordinates x = a, x = b.
 If n increases, the number of rectangles will increases and the width of rectangles will
decrease.
 The process of evaluating a definite integral by using the above definition is called
integration from first principles or integration by ab-initio method or integration as the
limit of a sum.

 Fundamental theorem of integral calculus states that if f(x) is a continuous function defined
x

on closed interval [a, b] and F(x) is integral of f(x) i.e., f ( )dx  F ( ),then
x
b b

a
b
x
 f ( )dx  ( F  ) x  F ( ) F ( )
a a
 Fundamental theorem of integral calculus is very useful as it gives us a method of calculating
the definite integral more easily, without calculating the limit of a sum.
4.4 Keywords
Definite Integral: The Definite Integral comprises extensive number of applications in
mathematics, the physical sciences and engineering.

Fundamental Theorem of Integral Calculus: It states that if f(x) is a continuous function defined
on closed interval [a, b] and F(x) is integral of f(x).

4.5 Review Questions

1. Elucidate the concept of Definite Integral as the Limit of a Sum. Give examples.

3
2. Evaluate the definite integral  x  3dx as limit of sums.
0
5

3. Evaluate the definite integral  1 xdx as limit of sums.
1
4
4. Evaluate the definite integral  3x  1dx as limit of sums.
2
2
2
5. Evaluate the definite integral  3x  2dx as limit of sums.
0
4
2

6. Evaluate the definite integral  2x x dx as limit of sums.
1
3
2
7. Evaluate the definite integral  x  2xdx as limit of sums.
1

66 LOVELY PROFESSIONAL UNIVERSITY