Page 48 - DMTH202_BASIC_MATHEMATICS_II

Richa Nandra, Lovely Professional University Unit 4: Definite Integral

Unit 4: Definite Integral Notes

CONTENTS
Objectives
Introduction

4.1 Definite Integral as the Limit of a Sum
4.2 Fundamental Theorem of Integral Calculus
4.3 Summary

4.4 Keywords
4.5 Review Questions
4.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand definite integral as the limit of a sum
 Discuss the fundamental theorem of integral calculus

Introduction

The Definite Integral comprises extensive number of applications in mathematics, the physical
sciences and engineering. The speculation and application of statistics, for instance, is based
greatly on the definite integral; via statistics, many conventionally non-mathematical regulations
have turn out to be greatly reliant on mathematical thoughts. Economics, sociology, psychology,
political science, geology, and many others specialized fields make use of calculus notions.

4.1 Definite Integral as the Limit of a Sum

Let f(x) be a continuous real valued function defined on the closed interval [a, b]. Divide the
interval [a,b] into n equal parts each of width h by points
a+h, a+2h, a + 3h, …, a+(n – 1) h.
h h   h  
b a


Then,
ntimes
b a

 nh  b a  .
h

n
Now the areas of inner rectangles are:
h
h
h
hf ( ),hf (a h ),hf (a  2 ),hf (a  3 ), ,hf (a n  1 ).
a


f
h

[ The breadth of inner rectangles is h and their heights are ( ), (f a f a h ), , (a n  1 )
respectively and area of rectangle is breadth  height]
LOVELY PROFESSIONAL UNIVERSITY 43