APPLIED

CALCULUSFOR ENGINEERING & SCIENCE STUDENTS

APPLIED

CALCULUSFOR ENGINEERING & SCIENCE STUDENTS

SAMSUDIN ABDULLAHRAHIMAH JUSOH @ AWANGEZZATUL FARHAIN AZMINORAZALIZA MOHD JAMILABDUL RAHMAN MOHD KASIMNOR IZZATI JAINI

PublisherUniversiti Malaysia Pahang

Kuantan2017

i

PREFACE

This module is designed for one-semester calculus course aimed at students majoring in science and engineering. A wide variety of topics, examples and problems are provided. This module can be used as an aided tool in teaching and learning multivariable calculus.

This module consists of four chapters. Students are introduced to polar coordinates and vectors in Chapter 1. This chapter includes the 3-dimensional graph sketching and polar curve, allowing for a source of rich illustrations and exercises. The next chapter concentrate on vector-valued functions including discussions on arc length, curvature and motion in space. Chapter 3 covers partial derivative involving functions with more than one variable. The final chapter covers multiple integrals and the applications.

This text also provides a complete set of teaching materials for courses in multivariable calculus that incorporates recent curricular and pedagogical developments in teaching and learning of calculus. An effort has been made to present the material clearly and at a level of sophistication appropriate for the audience. With the knowledge that any book can always be improved, we welcome corrections, constructive criticism, and suggestions from every reader.

Copyright Universiti Malaysia Pahang, 2017

First Published, 2017

Published By:Publisher

Universiti Malaysia Pahang,Lebuhraya Tun Razak,Pahang Darul Makmur.

Tel: 09-549 3320 Fax: 09-549 3381

Printing:Percetakan Muafakat Jaya Sdn. Bhd (105038-M)

6, Jalan Perdagangan 16, Taman Universiti Industrial Park,81300, Skudai Johor.

Tel: 07-520 6740 Fax: 07-520 6741

APPLIED CALCULUS FOR ENGINEERING & SCIENCE STUDENTS /SAMSUDIN BIN ABDULLAH RAHIMAH BINTI JUSOH @ AWANG EZZATUL FARHAIN BINTI AZMI DR. NORAZALIZA MOHD JAMIL DR. ABDUL RAHMAN BIN MOHD KASIM NOR IZZATI BINTI JAINI-NEW VERSIONISBN 978-967-2054-26-91. Calculus--Problems, exercises, etc. I. Rahimah Jusoh @ AwangII. Ezzatul Farhain Azmi. III. Norazaliza Mohd Jamil, Dr.IV. Abdul Rahman Mohd. Kasim, Dr. V. Nor Izzati Jaini.VII. Title.515.076

Samsudin Abdullah

All right reserved.Apart from fair dealing for the purpose of study, research,

criticism or review, as permitted under the Copyright Act, no part of thisbook may be reproduced, strored in a retrieval system, or transmited, in any form or

by any means, electronic, mechanical, photocopying, recording or otherwisewithout the prior written permission of the publisher. Enquiries to be made

to the author and the publisher Penerbit Universiti Malaysia Pahang,Lebuhraya Tun Razak, 26300 Gambang, Kuantan, Pahang Darul Makmur.

Negotiation is subject to royalty arrangement or honorarium.

i

PREFACE

This module is designed for one-semester calculus course aimed at students majoring in science and engineering. A wide variety of topics, examples and problems are provided. This module can be used as an aided tool in teaching and learning multivariable calculus.

This module consists of four chapters. Students are introduced to polar coordinates and vectors in Chapter 1. This chapter includes the 3-dimensional graph sketching and polar curve, allowing for a source of rich illustrations and exercises. The next chapter concentrate on vector-valued functions including discussions on arc length, curvature and motion in space. Chapter 3 covers partial derivative involving functions with more than one variable. The final chapter covers multiple integrals and the applications.

This text also provides a complete set of teaching materials for courses in multivariable calculus that incorporates recent curricular and pedagogical developments in teaching and learning of calculus. An effort has been made to present the material clearly and at a level of sophistication appropriate for the audience. With the knowledge that any book can always be improved, we welcome corrections, constructive criticism, and suggestions from every reader.

Copyright Universiti Malaysia Pahang, 2017

First Published, 2017

Published By:Publisher

Universiti Malaysia Pahang,Lebuhraya Tun Razak,Pahang Darul Makmur.

Tel: 09-549 3320 Fax: 09-549 3381

Printing:Percetakan Muafakat Jaya Sdn. Bhd (105038-M)

6, Jalan Perdagangan 16, Taman Universiti Industrial Park,81300, Skudai Johor.

Tel: 07-520 6740 Fax: 07-520 6741

APPLIED CALCULUS FOR ENGINEERING & SCIENCE STUDENTS /SAMSUDIN BIN ABDULLAH RAHIMAH BINTI JUSOH @ AWANG EZZATUL FARHAIN BINTI AZMI DR. NORAZALIZA MOHD JAMIL DR. ABDUL RAHMAN BIN MOHD KASIM NOR IZZATI BINTI JAINI-NEW VERSIONISBN 978-967-2054-26-91. Calculus--Problems, exercises, etc. I. Rahimah Jusoh @ AwangII. Ezzatul Farhain Azmi. III. Norazaliza Mohd Jamil, Dr.IV. Abdul Rahman Mohd. Kasim, Dr. V. Nor Izzati Jaini.VII. Title.515.076

Samsudin Abdullah

All right reserved.Apart from fair dealing for the purpose of study, research,

criticism or review, as permitted under the Copyright Act, no part of thisbook may be reproduced, strored in a retrieval system, or transmited, in any form or

by any means, electronic, mechanical, photocopying, recording or otherwisewithout the prior written permission of the publisher. Enquiries to be made

to the author and the publisher Penerbit Universiti Malaysia Pahang,Lebuhraya Tun Razak, 26300 Gambang, Kuantan, Pahang Darul Makmur.

Negotiation is subject to royalty arrangement or honorarium.

ii

CHAPTER 1 : POLAR COORDINATES & VECTORS 1

1.0 Introduction ……………………………………….………..………………................ 3

1.1 Parametric equations ……...………………………………………………………… 7

EXERCISES 1.1 ……………………………………………………………………...14

1.2 Three Dimensional Coordinate Systems ………………………………….............17

1.2.1 Line, Plane and Space …………..……………………………………...….. 20

1.2.2 Surfaces (Planes) ……………………….…………………………….......... 21

1.2.3 Distance in Three Dimensional System …………………………….......... 24

EXERCISES 1.2 ……………………………………………………………………. 25

1.3 Cylindrical and Quadric Surfaces ………………………………………………… 29

1.3.1 Level Curves ………………………………………………………………….29

1.3.2 Cylindrical Surfaces …………………………………………………………31

1.3.3 Paraboloids ……………………………………………………………......... 35

1.3.4 Spheres …...…………. …………………………………………………...... 38

1.3.5 Ellipsoids………………………………………………………………………43

1.3.6 Cones …………………………………………………………………………44

EXERCISES 1.3 …………………………………………………………………… 45

1.4 Lines and Planes ……………………………………………………………………49

1.4.1 Equation of Lines …………………………………………………………….49

1.4.2 Characteristics of Lines ……….…………………………………………….55

1.4.3 Equation of Planes ……………………………………………………..........58

1.4.4 Intersecting Planes ……………………………………………………..........64

1.4.5 Distance between Two Planes …………………………………………......65

EXERCISES 1.4 …………………………………………………………………….67

1.5 Polar Coordinates …………………………………………………………………..72

1.5.1 Relationship between Polar and Rectangular Coordinates …………….75

1.5.2 Graphing Polar Curve.………………………………………………………79

1.5.3 Symmetric Test ………………………………………………………………95

EXERCISES 1.5 …………………………………………………………………...100

TUTORIAL 1 : Polar Coordinates & Vectors ……………………………………………...105

TABLE OF CONTENTS

iii

CHAPTER 2 : VECTOR-VALUED FUNCTIONS 115

2.0 Introduction …………………………………………………………………………...117

2.1 Vector Functions and Space Curves……………………………………………….118

2.1.1 Domain and Graph of Selected Functions ……………………………....118

2.1.2 Operational Properties of Vector Functions ……………………………..126

EXERCISES 2.1 ………………………………………………………………..........128

2.2 Derivatives and Integrals of Vector Functions ……………………………………130

2.2.1 Derivatives of Vector Functions ..…………………………………………...130

2.2.2 Tangent Line and Tangent Vector …….……………………………………131

2.2.3 Differentiation Rules …………………………………………………............136

2.2.4 Integration of Vector Functions ……………………………………………..138

EXERCISES 2.2 ……………………………………………………………………..140

2.3 Arc Length and Curvature …………………………………………………………..142

2.3.1 Unit Tangent, Unit Normal and Binormal Vectors ...……………………....142

2.3.2 Arc Length …………………………………………………............................147

2.3.3 Curvatures ………………………………………………………………….....148

EXERCISES 2.3 ……………………………………………………………………..151

2.4 Motion in Space : Velocity and Acceleration ……………………………………...153

2.4.1 Normal and Tangential Components of Acceleration ………………….....156

EXERCISES 2.4 ……………………………………………………………………...158

TUTORIAL 2 : Vector-valued Functions ………………………………………………….....160

CHAPTER 3 : PARTIAL DERIVATIVES 163

3.0 Introduction …………………………………………………………………………..165

3.1 Functions of Two or More Variables…………………………………………….....165

3.1.1 Graphing Function of Two Independent Variables ……………………….169

EXERCISE 3.1 ……………………………………………………………………….171

3.2 Partial Derivatives …………………………………................................................172

3.2.1 Partial Derivative as A Slope ...……………………………………………...175

3.2.2 Partial Derivative as Rate of Change …….…………………………….......177

3.2.3 Higher Order Partial Derivative ……………………………………………..178

3.2.4 Geometric Interpretation of Second Order Partial Derivatives …………..179

EXERCISES 3.2 …………………………………………………………………......181

3.3 The Chain Rule ………………………………………………………………………184

iv

EXERCISES 3.3 ………………………………………………………………………190

3.4 Implicit Partial Differentiation ………………………………………………………..192

EXERCISES 3.4 ………………………………………………………………………199

3.5 Extrema of Functions of Two Variables …………………………………………….200

EXERCISES 3.5 ……………………………………………………………………....205

TUTORIAL 3 : Partial Derivatives ……………………………………………………………..206

CHAPTER 4 : MULTIPLE INTEGRALS 211

4.0 Introduction…………………………………………………………………………….213

4.1 Area in Polar Coordinates……………………………………………………………213

EXERSICES 4.1 ……………………………………………………………………....223

4.2 Double Integrals over Rectangular Region………………………………………....224

4.2.1 Volume under a Surface ………………………………………………….......224

4.2.2 Fubini’s Theorem ……………………………………………………………...226

4.2.3 Properties of Double Integrals ……………………………………….............229

EXERCISES 4.2 …………………………………………………………………….. .230

4.3 Double Integrals over Nonrectangular Region………………………………….....232

4.3.1 Area and Volume ..…………………………………………………………....233

4.3.2 Reverse The Order of Integration …….……………………………..............237

EXERCISES 4.3 ……………………………………………………………………..239

4.4 Double Integrals in Polar Coordinates …………………….……………………....241

EXERCISES 4.4 ……………………………………………………………………..250

4.5 Triple Integrals ………………………………………………………………………….252

4.5.1 Properties of Triple Integrals ……………………………..………………….252

4.5.2 Evaluating Triple Integrals Over More General Regions………………….253

4.5.3 Volume Calculated as A Triple Integral……………………………………..254

EXERCISES 4.5 ……………………………………………………………………...256

4.6 Triple Integrals in Cylindrical and Spherical Coordinates…………………………..257

4.6.1 Cylindrical Coordinates……………………………………………….............257

EXERCISES 4.6.1 ……………………………………………………………………261

4.6.2 Triple Integrals in Cylindrical Coordinates………………………………….262

EXERCISES 4.6.2 ……………………………………………………………………266

iii

CHAPTER 2 : VECTOR-VALUED FUNCTIONS 115

2.0 Introduction …………………………………………………………………………...117

2.1 Vector Functions and Space Curves……………………………………………….118

2.1.1 Domain and Graph of Selected Functions ……………………………....118

2.1.2 Operational Properties of Vector Functions ……………………………..126

EXERCISES 2.1 ………………………………………………………………..........128

2.2 Derivatives and Integrals of Vector Functions ……………………………………130

2.2.1 Derivatives of Vector Functions ..…………………………………………...130

2.2.2 Tangent Line and Tangent Vector …….……………………………………131

2.2.3 Differentiation Rules …………………………………………………............136

2.2.4 Integration of Vector Functions ……………………………………………..138

EXERCISES 2.2 ……………………………………………………………………..140

2.3 Arc Length and Curvature …………………………………………………………..142

2.3.1 Unit Tangent, Unit Normal and Binormal Vectors ...……………………....142

2.3.2 Arc Length …………………………………………………............................147

2.3.3 Curvatures ………………………………………………………………….....148

EXERCISES 2.3 ……………………………………………………………………..151

2.4 Motion in Space : Velocity and Acceleration ……………………………………...153

2.4.1 Normal and Tangential Components of Acceleration ………………….....156

EXERCISES 2.4 ……………………………………………………………………...158

TUTORIAL 2 : Vector-valued Functions ………………………………………………….....160

CHAPTER 3 : PARTIAL DERIVATIVES 163

3.0 Introduction …………………………………………………………………………..165

3.1 Functions of Two or More Variables…………………………………………….....165

3.1.1 Graphing Function of Two Independent Variables ……………………….169

EXERCISE 3.1 ……………………………………………………………………….171

3.2 Partial Derivatives …………………………………................................................172

3.2.1 Partial Derivative as A Slope ...……………………………………………...175

3.2.2 Partial Derivative as Rate of Change …….…………………………….......177

3.2.3 Higher Order Partial Derivative ……………………………………………..178

3.2.4 Geometric Interpretation of Second Order Partial Derivatives …………..179

EXERCISES 3.2 …………………………………………………………………......181

3.3 The Chain Rule ………………………………………………………………………184

iv

EXERCISES 3.3 ………………………………………………………………………190

3.4 Implicit Partial Differentiation ………………………………………………………..192

EXERCISES 3.4 ………………………………………………………………………199

3.5 Extrema of Functions of Two Variables …………………………………………….200

EXERCISES 3.5 ……………………………………………………………………....205

TUTORIAL 3 : Partial Derivatives ……………………………………………………………..206

CHAPTER 4 : MULTIPLE INTEGRALS 211

4.0 Introduction…………………………………………………………………………….213

4.1 Area in Polar Coordinates……………………………………………………………213

EXERSICES 4.1 ……………………………………………………………………....223

4.2 Double Integrals over Rectangular Region………………………………………....224

4.2.1 Volume under a Surface ………………………………………………….......224

4.2.2 Fubini’s Theorem ……………………………………………………………...226

4.2.3 Properties of Double Integrals ……………………………………….............229

EXERCISES 4.2 …………………………………………………………………….. .230

4.3 Double Integrals over Nonrectangular Region………………………………….....232

4.3.1 Area and Volume ..…………………………………………………………....233

4.3.2 Reverse The Order of Integration …….……………………………..............237

EXERCISES 4.3 ……………………………………………………………………..239

4.4 Double Integrals in Polar Coordinates …………………….……………………....241

EXERCISES 4.4 ……………………………………………………………………..250

4.5 Triple Integrals ………………………………………………………………………….252

4.5.1 Properties of Triple Integrals ……………………………..………………….252

4.5.2 Evaluating Triple Integrals Over More General Regions………………….253

4.5.3 Volume Calculated as A Triple Integral……………………………………..254

EXERCISES 4.5 ……………………………………………………………………...256

4.6 Triple Integrals in Cylindrical and Spherical Coordinates…………………………..257

4.6.1 Cylindrical Coordinates……………………………………………….............257

EXERCISES 4.6.1 ……………………………………………………………………261

4.6.2 Triple Integrals in Cylindrical Coordinates………………………………….262

EXERCISES 4.6.2 ……………………………………………………………………266

v

4.6.3 Spherical Coordinates ……………………………………………………….268

EXERCISES 4.6.3 …………………………………………………………………...271

4.6.4 Triple Integrals in Spherical Coordinates ………………………………….272

EXERCISES 4.6.4 …………………………………………………………………...275

4.7 Applications of Multiple Integrals ………………………………………………........277

4.7.1 Surface Area ………………………………………………………………….277

4.7.2 Mass …………………………………………………………………...............281

4.7.3 Center of Gravity and Centroid of Lamina …………………………………282

4.7.4 Center of Gravity and Centroid of Solid …………………………………....284

EXERCISES 4.7 ……………………………………………………………………..289

TUTORIAL 4 : Multiple Integrals ………………………………………………………………291

APPENDICES ………………………………………………………………………………......299

Chapter 1: Polar Coordinates and Vectors 1

Universiti Malaysia Pahang BUM2123 Applied Calculus

“Success is the sum of small efforts, repeated day in and day out.”

- Robert Collier

Polar Coordinates and Vectors

Chapter 3: Partial Derivatives 163

Universiti Malaysia Pahang BUM2123 Applied Calculus

“The greater the obstacle, the more glory in overcoming it.”

- Moliere

Partial Derivatives