Investagacion Modelos Matematicos Integracion

Mathematics Learning Centre

Introduction to Integration Part 2: The Definite Integral Mary Barnes

c 1999

University of Sydney

Contents
1 Introduction 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 7 7 8 12 14

2 Finding Areas 3 Areas Under Curves 3.1 3.2 What is the point of allthis?

Note about summation notation . . . . . . . . . . . . . . . . . . . . . . . .

4 The Definition of the Definite Integral 4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 The Fundamental Theorem of the Calculus 6 Properties of the Definite Integral 7 Some Common Misunderstandings 7.1 7.2

Arbitrary constants . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 14 Dummy variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15 19 21 23

8 Another Look at Areas 9 The Area Between Two Curves 10 Other Applications of the Definite Integral 11 Solutions to Exercises

Mathematics Learning Centre, University of Sydney

1

1

Introduction

This unit deals with the definite integral. It explains how it isdefined, how it is calculated and some of the ways in which it is used. We shall assume that you are already familiar with the process of finding indefinite integrals or primitive functions (sometimes called anti-differentiation) and are able to ‘antidifferentiate’ a range of elementary functions. If you are not, you should work through Introduction to Integration Part I: Anti-Differentiation, and make sureyou have mastered the ideas in it before you begin work on this unit.

1.1

Objectives

By the time you have worked through this unit you should: • Be familiar with the definition of the definite integral as the limit of a sum; • Understand the rule for calculating definite integrals; • Know the statement of the Fundamental Theorem of the Calculus and understand what it means; • Be able to usedefinite integrals to find areas such as the area between a curve and the x-axis and the area between two curves; • Understand that definite integrals can also be used in other situations where the quantity required can be expressed as the limit of a sum.

Mathematics Learning Centre, University of Sydney

2

2

Finding Areas

Areas of plane (i.e. flat!) figures are fairly easy to calculateif they are bounded by straight lines. The area of a rectangle is clearly the length times the breadth. The area of a right-angled triangle can be seen to be half the area of a rectangle (see the diagram) and so is half the base times the height.

Area of rectangle =length × breadth

Area of triangle = 1 area of rectangle 2 = 1 length × breadth 2
A

The areas of other triangles can befound by expressing them as the sum or the difference of the areas of right angled triangles, and from this it is clear that for any triangle this area is half the base times the height.

A

B

D

C

B

C

D

Area of ABC = area of ABD + area of ACD Using this, we can find the area of any figure bounded by straight lines, by dividing it up into triangles (as shown).

Area of ABC = areaof ABD − area of ACD

Areas bounded by curved lines are a much more difficult problem, however. In fact, although we all feel we know intuitively what we mean by the area of a curvilinear figure, it is actually quite difficult to define precisely. The area of a figure is quantified by asking ‘how many units of area would be needed to cover it?’ We need to have some unit of area in mind (e.g. one squarecentimetre or one square millimetre) and imagine trying to cover the figure with little square tiles. We can also imagine cutting these tiles in halves, quarters etc. In this way a rectangle, and hence any figure bounded by straight lines, can be dealt with, but a curvilinear figure can never be covered exactly. We are therefore forced to rely on the notion of limit in order to define areas of...