Integral Calculus Course Notes: Areas, Volumes, and Techniques

Integral Calculus: Mathematics 103
Notes by Leah Edelstein-Keshet: All rights reserved
University of British Columbia
January 2, 2010

ii Leah Edelstein-Keshet

Contents
Preface xvii
1 Areas, volumes and simple sums 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Areas of simple shapes . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Example 1: Finding the area of a polygon using triangles:
a “dissection” method . . . . . . . . . . . . . . . . . . . 3
1.2.2 Example 2: How Archimedes discovered the area of a
circle: dissect and “take a limit” . . . . . . . . . . . . . . 4
1.3 Simple volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Example 3: The Tower of Hanoi: a tower of disks . . . . 8
1.4 Summations and the “Sigma” notation . . . . . . . . . . . . . . . . . 9
1.4.1 Manipulations of sums . . . . . . . . . . . . . . . . . . . 10
1.5 Summation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Example 3, revisited: Volume of a Tower of Hanoi . . . . 12
1.6 Summing the geometric series . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Prelude to infinite series . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7.1 The infinite geometric series . . . . . . . . . . . . . . . . 15
1.7.2 Example: A geometric series that converges. . . . . . . . 16
1.7.3 Example: A geometric series that diverges . . . . . . . . 16
1.8 Application of geometric series to the branching structure of the lungs 16
1.8.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8.2 A simple geometric rule . . . . . . . . . . . . . . . . . . 19
1.8.3 Total number of segments . . . . . . . . . . . . . . . . . 20
1.8.4 Total volume of airways in the lung . . . . . . . . . . . . 20
1.8.5 Total surface area of the lung branches . . . . . . . . . . 21
1.8.6 Summary of predictions for specific parameter values . . 22
1.8.7 Exploring the problem numerically . . . . . . . . . . . . 23
1.8.8 For further independent study . . . . . . . . . . . . . . . 23
1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Areas 27
2.1 Areas in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Computing the area under a curve by rectangular strips . . . . . . . . 29
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iv Contents
2.2.1 First approach: Numerical integration using a spreadsheet 29
2.2.2 Second approach: Analytic computation using Riemann
sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 The area of a leaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Area under an exponential curve . . . . . . . . . . . . . . . . . . . . 35
2.5 Extensions and other examples . . . . . . . . . . . . . . . . . . . . . 36
2.6 The definite integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 The area as a function . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 The Fundamental Theorem of Calculus 43
3.1 The definite integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Properties of the definite integral . . . . . . . . . . . . . . . . . . . . 44
3.3 The area as a function . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . 47
3.4.1 Fundamental theorem of calculus: Part I . . . . . . . . . 47
3.4.2 Example: an antiderivative . . . . . . . . . . . . . . . . . 47
3.4.3 Fundamental theorem of calculus: Part II . . . . . . . . . 48
3.5 Review of derivatives (and antiderivatives) . . . . . . . . . . . . . . . 49
3.6 Examples: Computing areas with the Fundamental Theorem of Calculus 50
3.6.1 Example 1: The area under a polynomial . . . . . . . . . 50
3.6.2 Example 2: Simple areas . . . . . . . . . . . . . . . . . 50
3.6.3 Example 3: The area between two curves . . . . . . . . . 52
3.6.4 Example 4: Area of land . . . . . . . . . . . . . . . . . . 53
3.7 Qualitative ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7.1 Example: sketching A(x). . . . . . . . . . . . . . . . . 56
3.8 Some fine print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8.1 Function unbounded I . . . . . . . . . . . . . . . . . . . 57
3.8.2 Function unbounded II . . . . . . . . . . . . . . . . . . . 57
3.8.3 Example: Function discontinuous or with distinct parts . 57
3.8.4 Function undefined . . . . . . . . . . . . . . . . . . . . 57
3.8.5 Infinite domain (“improper integral”) . . . . . . . . . . . 58
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Applications of the definite integral to velocities and rates 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Displacement, velocity and acceleration . . . . . . . . . . . . . . . . 62
4.2.1 Geometric interpretations . . . . . . . . . . . . . . . . . 62
4.2.2 Displacement for uniform motion . . . . . . . . . . . . . 63
4.2.3 Uniformly accelerated motion . . . . . . . . . . . . . . . 63
4.2.4 Non-constant acceleration and terminal velocity . . . . . 64
4.3 From rates of change to total change . . . . . . . . . . . . . . . . . . 66
4.3.1 Tree growth rates . . . . . . . . . . . . . . . . . . . . . . 68

Contents v
4.3.2 Radius of a tree trunk . . . . . . . . . . . . . . . . . . . 68
4.3.3 Birth rates and total births . . . . . . . . . . . . . . . . . 71
4.4 Production and removal . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Present value of a continuous income stream . . . . . . . . . . . . . . 74
4.6 Average value of a function . . . . . . . . . . . . . . . . . . . . . . . 76
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Applications of the definite integral to calculating volume, mass, and length 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Mass distributions in one dimension . . . . . . . . . . . . . . . . . . 82
5.2.1 A discrete distribution: total mass of beads on a wire . . . 82
5.2.2 A continuous distribution: mass density and total mass . . 82
5.2.3 Example: Actin density inside a cell . . . . . . . . . . . 84
5.3 Mass distribution and the center of mass . . . . . . . . . . . . . . . . 85
5.3.1 Center of mass of a discrete distribution . . . . . . . . . . 85
5.3.2 Center of mass of a continuous distribution . . . . . . . . 85
5.3.3 Example: Center of mass vs average mass density . . . . 86
5.3.4 Physical interpretation of the center of mass . . . . . . . 87
5.4 Miscellaneous examples and related problems . . . . . . . . . . . . . 87
5.4.1 Example: A glucose density gradient . . . . . . . . . . . 87
5.4.2 Example: A circular colony of bacteria . . . . . . . . . . 89
5.5 Volumes of solids of revolution . . . . . . . . . . . . . . . . . . . . . 90
5.5.1 Volumes of cylinders and shells . . . . . . . . . . . . . . 90
5.5.2 Computing the Volumes . . . . . . . . . . . . . . . . . . 91
5.6 Length of a curve: Arc length . . . . . . . . . . . . . . . . . . . . . . 96
5.6.1 How the alligator gets its smile . . . . . . . . . . . . . . 99
5.6.2 References . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Techniques of Integration 107
6.1 Differential notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Antidifferentiation and indefinite integrals . . . . . . . . . . . . . . . 110
6.2.1 Integrals of derivatives . . . . . . . . . . . . . . . . . . . 111
6.3 Simple substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3.1 Example: Simple substitution . . . . . . . . . . . . . . . 112
6.3.2 How to handle endpoints . . . . . . . . . . . . . . . . . 113
6.3.3 Examples: Substitution type integrals . . . . . . . . . . . 113
6.3.4 When simple substitution fails . . . . . . . . . . . . . . . 115
6.3.5 Checking your answer . . . . . . . . . . . . . . . . . . . 116
6.4 More substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.1 Example: perfect square in denominator . . . . . . . . . 116
6.4.2 Example: completing the square . . . . . . . . . . . . . . 117
6.4.3 Example: factoring the denominator . . . . . . . . . . . 117
6.5 Trigonometric substitutions . . . . . . . . . . . . . . . . . . . . . . . 118
6.5.1 Example: simple trigonometric substitution . . . . . . . . 118
6.5.2 Example: using trigonometric identities (1) . . . . . . . . 119