MA141 Chapters 4 and 5 versionf16 - NCSU Calculus Volume I...

NCSU Calculus, Volume I
John E. Franke, John R. Griggs and Larry K. Norris
August 10, 2016

Calculus I 2 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
Last update: August 10, 2016

Contents
4 Integration 5
4.1 Areas and Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Method of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 61
5 Applications of Integration 63
5.1 Area Between Two Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Volumes of Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.1 Disk Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.3 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 86
5.2.4 Washer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.6 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 95
5.2.7 Cylindrical Shell Method . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.9 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 100
Index 101
Calculus I 3 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
Last update: August 10, 2016

CONTENTS
Calculus I 4 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
Last update: August 10, 2016

Chapter 4
Integration
Comments and suggestions are welcome!
In this chapter we make a transition from indefinite integration to definite integration. To
do this we take a side trip to the concept of the area under a curve . The purpose of this
discussion is to motivate and connect the previously developed concepts in Chapter 3 to the
new ones we develop in Section 4.2. The Fundamental Theorem of Calculus provides the
connection between these concepts. The last sections are devoted to the first two of the many
techniques of integration covered in calculus.
4.1 Areas and Riemann Sums
The development of the definite integral involves quite a few sums. To compactly express
these sums, we will use summation notation . If we have the sum of n numbers
a
1
+ a
2
+ a
3
+ · · · + a
n
the notation
n X
i =1
a
i
can be used to represent this sum succinctly. That is
n X
i =1
a
i
= a
1
+ a
2
+ a
3
+ · · · + a
n
The Greek capital letter sigma Σ indicates a sum and the symbol a
i
represents the i th term.
The variable i is called the index of summation . The values 1 and n indicate the beginning
and ending values for i - taking on all consecutive integers between 1 and n inclusive. Letters
other than i can be used and not all sums begin with i = 1.
Example 1. Evaluate the sum
5
X
i =1
4 i
2
Calculus I 5 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
Last update: August 10, 2016

4.1. AREAS AND RIEMANN SUMS
Solution:
5 X
i =1
4 i
2
= 4(1)
2
+ 4(2)
2
+ 4(3)
2
+ 4(4)
2
+ 4(5)
2
= 4 + 16 + 36 + 64 + 100
= 220
The nature of these sums allows for some algebraic simplifications. The following properties
can be proved using the expanded form of each sum.
Theorem 1. Algebra of Summation
Let n be a natural number, a
i
and b
i
be real numbers for i from 1 to n , and c ∈ R ,
then
1 .