Page 269 - DMTH401_REAL ANALYSIS

Richa Nandra, Lovely Professional University Unit 21: Properties of Integrals

Unit 21: Properties of Integrals Notes

CONTENTS
Objectives
Introduction
21.1 Properties of Riemann Integral
21.2 Summary
21.3 Keyword

21.4 Review Questions
21.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Identify the properties of the integral and
 Use them to find the Riemann Stieltjes integral of functions

Introduction

In last unit you have studied about Riemann integral. In this unit, we are going to see the
properties of Riemann Stieltjes integral.

21.1 Properties of Riemann Integral

As you were introduced to some methods which enabled you to associate with each integrable
b
function f defined on [a,b], a unique real number called the integral f(x) dx in the sense of
ò
a
Riemann. A method of computing this integral as a limit of a sum was explained. All this leads
us to consider some nice properties which are presented as follows:
Property 1: If f and g are integrable on [a, b] and if
f(x) g(x) " x [a,b],
£
Î
then

b b
ò f(x) dx £ ò g(x) dx

Proof: Define a function h: [a,b]  R as
h = g – f .
Since f and g are integrable on [a, b], therefore, the difference h is also integrable on [a, b].
Since
f(x) £ g(x)  g(x) – f(x)  0,

therefore h(x)  0 for all x E [a,b].

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