Page 244 - DMTH401_REAL ANALYSIS

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Notes Unit 20: The Riemann Integration

CONTENTS
Objectives
Introduction
20.1 Riemann Integration
20.2 Riemann Integrable Functions

20.3 Algebra of Integrable Functions
20.4 Computing an Integral
20.5 Summary
20.6 Keywords
20.7 Review Questions
20.8 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the Riemann Integral of a function
 Derive the conditions of Integrability and determine the class of functions which are
always integrable
 Discuss the algebra of integrable functions

 Compute the integral as a limit of a sum
Introduction

You are quite familiar with the words ‘differentiation’ and distinguishing ‘integration’. You
know that in ordinary language, differentiation refers to separating things while integration
means putting things together. In Mathematics, particularly in Calculus and Analysis,
differentiation and integration are considered as some kind of operations on functions. You
have used these operations in our study of Calculus.
There are essentially two ways of describing the operation of integration. One way is to view it
as the inverse operation of differentiation. The other way is to treat it as some sort of limit of a
sum.
The first view gives rise to an integral which is the result of reversing the process of
differentiation. This is the view which was generally considered during the eighteenth century.
Accordingly, the method is to obtain, from a given function, another function which has the first
function as its derivative. This second function, if it be obtained, is called the indefinite integral
of the first function. This is also called the 'primitive' or anti-derivative of the first function.
Thus, the integral of a function f(x) is obtained by finding an anti-derivative or primitive function
F(x) show that F’(x) = f (x). The indefinite integral of f(x), is symbolized by the notation f(x) dx.

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