Calculus II
MATH 152 Course Notes
Department. of Mathematics, SFU
Spring 2023
,Copyright © 2023 Veselin Jungic and Jamie Mulholland, SFU
S ELF P UBLISHED
http://www.math.sfu.ca
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License (the
“License”). You may not use this document except in compliance with the License. You may obtain
a copy of the License at http://creativecommons.org/licenses/by-nc-sa/4.0/. Unless
required by applicable law or agreed to in writing, software distributed under the License is dis-
tributed on an “AS IS ” BASIS , WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either
express or implied. See the License for the specific language governing permissions and limitations
under the License.
First printing, August 2006
, Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I Part One: Introduction to the Integral
1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 Areas and Distances 14
1.2 The Definite Integral 19
1.3 The Fundamental Theorem of Calculus 28
1.4 The Net Change Theorem 34
1.5 The Substitution Rule 39
2 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1 Areas Between Curves 48
2.2 Areas in Polar Coordinates 52
2.3 Volumes 55
2.4 Volumes by Cylindrical Shells 62
II Part Two: Integration Techniques and Applications
3 Techniques of Integration and Applications . . . . . . . . . . . . . . . . . . . . 69
3.1 Integration By Parts 70
3.2 Trigonometric Integrals 76
3.3 Trigonometric Substitutions 80
, 3.4 Integration of Rational Functions by Partial Fractions 82
3.5 Strategy for Integration 89
3.6 Approximate Integration 95
3.7 Improper Integrals 103
4 Further Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.1 Arc Length 110
4.2 Area of a Surface of Revolution 114
4.3 Calculus with Parametric Curves 118
III Part Three: Sequences and Series
5 Infinite Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1 Sequences 126
5.2 Series 133
5.3 The Integral Test and Estimates of Sums 139
5.4 The Comparison Test 143
5.5 Alternating Series 147
5.6 Absolute Convergence and the Ratio and Root Test 151
5.7 Strategy for Testing Series 156
5.8 Power Series 160
5.9 Representation of Functions as Power Series 164
5.10 Taylor and Maclaurin Series 167
5.11 Applications of Taylor Polynomials 175
IV Part Four: Differential Equations
6 A First Look at Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.1 Modeling with Differential Equations, Directions Fields 182
6.2 Separable Equations 187
6.3 Models for Population Growth 194
V Exam Preparation
7 Review Materials for Exam Preparation . . . . . . . . . . . . . . . . . . . . . . . . 199
7.1 Midterm 1 Review Package 200
7.2 Midterm 2 Review Package 207
7.3 Final Exam Practice Questions 216
VI Appendix
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Articles 223
Books 223
MATH 152 Course Notes
Department. of Mathematics, SFU
Spring 2023
,Copyright © 2023 Veselin Jungic and Jamie Mulholland, SFU
S ELF P UBLISHED
http://www.math.sfu.ca
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License (the
“License”). You may not use this document except in compliance with the License. You may obtain
a copy of the License at http://creativecommons.org/licenses/by-nc-sa/4.0/. Unless
required by applicable law or agreed to in writing, software distributed under the License is dis-
tributed on an “AS IS ” BASIS , WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either
express or implied. See the License for the specific language governing permissions and limitations
under the License.
First printing, August 2006
, Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I Part One: Introduction to the Integral
1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 Areas and Distances 14
1.2 The Definite Integral 19
1.3 The Fundamental Theorem of Calculus 28
1.4 The Net Change Theorem 34
1.5 The Substitution Rule 39
2 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1 Areas Between Curves 48
2.2 Areas in Polar Coordinates 52
2.3 Volumes 55
2.4 Volumes by Cylindrical Shells 62
II Part Two: Integration Techniques and Applications
3 Techniques of Integration and Applications . . . . . . . . . . . . . . . . . . . . 69
3.1 Integration By Parts 70
3.2 Trigonometric Integrals 76
3.3 Trigonometric Substitutions 80
, 3.4 Integration of Rational Functions by Partial Fractions 82
3.5 Strategy for Integration 89
3.6 Approximate Integration 95
3.7 Improper Integrals 103
4 Further Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.1 Arc Length 110
4.2 Area of a Surface of Revolution 114
4.3 Calculus with Parametric Curves 118
III Part Three: Sequences and Series
5 Infinite Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1 Sequences 126
5.2 Series 133
5.3 The Integral Test and Estimates of Sums 139
5.4 The Comparison Test 143
5.5 Alternating Series 147
5.6 Absolute Convergence and the Ratio and Root Test 151
5.7 Strategy for Testing Series 156
5.8 Power Series 160
5.9 Representation of Functions as Power Series 164
5.10 Taylor and Maclaurin Series 167
5.11 Applications of Taylor Polynomials 175
IV Part Four: Differential Equations
6 A First Look at Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.1 Modeling with Differential Equations, Directions Fields 182
6.2 Separable Equations 187
6.3 Models for Population Growth 194
V Exam Preparation
7 Review Materials for Exam Preparation . . . . . . . . . . . . . . . . . . . . . . . . 199
7.1 Midterm 1 Review Package 200
7.2 Midterm 2 Review Package 207
7.3 Final Exam Practice Questions 216
VI Appendix
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Articles 223
Books 223
