This textbook offers an extensive list of completely solved problems in mathematical analysis. This second of three volumes covers definite, improper and multidimensional integrals, functions of several variables, differential equations, and more. The series contains the material corresponding to the first three or four semesters of a course in Mathematical Analysis. Based on the author's years of teaching experience, this work stands out by providing detailed solutions (often several pages long) to the problems. The basic premise of the book is that no topic should be left unexplained, and no question that could realistically arise while studying the solutions should remain unanswered.

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Problem Books in Mathematics

Series Editor:

Peter Winkler

Department of Mathematics

Dartmouth College

Hanover, NH 03755

USA

More information about this series at http://www.springer.com/series/714

Tomasz Rado ˙

zycki

Solving Problems

in Mathematical Analysis,

Part II

Definite, Improper and Multidimensional

Integrals, Functions of Several Variables

and Differential Equations

Tomasz Rado˙

zycki

Faculty of Mathematics and Natural

Sciences, College of Sciences

Cardinal Stefan Wyszy´

nski University

War s a w, P o l a n d

Scientific review for the Polish edition: Jerzy Jacek Wojtkiewicz

Based on a translation from the Polish language edition: "Rozwi ˛ azujemy zadania z analizy

matematycznej" cz˛ e´

s´

c 2 by Tomasz Rado˙

zycki Copyright ©WYDAWNICTWO O ´

SWIA-

TOWE "FOSZE" 2013 All Rights Reserved.

ISSN 0941-3502 ISSN 2197-8506 (electronic)

Problem Books in Mathematics

ISBN 978-3-030-36847-0 ISBN 978-3-030-36848-7 (eBook)

https://doi.org/10.1007/978-3-030-36848- 7

Mathematics Subject Classification: 00-01, 00A07, 34-XX, 34A25, 34K28, 53A04, 26Bxx

© Springer Nature Switzerland AG 2020

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of

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Preface

The second part of this book series covers, roughly speaking, the material of the

second semester course of mathematical analysis held by university departments of

sciences. I tried to preserve the character of the first part, presenting very detailed

solutions of all problems, without abbreviations or understatements. In contrast

to authors of typical problem sets, my goal was not to provide the reader many

examples for his own work in which only final results would be given (although

such problems are also provided at the ends of chapters), but to clarify all steps of

the solution to a given problem: from choosing the method, through its technical

implementation to the final result.

A priority for me was the simplicity of argumentation although I am aware that

the price for this is sometimes the loss of full precision. Concerned about the volume

of the book and knowing that it is intended for students who have already passed

the first semester of mathematical analysis, I let myself sometimes—in contrast to

the first part—skip the details of very elementary transformations. The inspiration

that guided me to prepare this particular set of problems is given in the preface of

the first part.

The theoretical summaries placed at the beginning of each chapter only serve to

collect the theorems and formulas that will be used. Theory of the mathematical

analysis should be learned from other textbooks or lectures. The problems contained

in this book can also be studied without reading these theoretical summaries since

all necessary notions are repeated informally in the solution where applicable.

Warsaw, Poland Tomasz Rado ˙

zycki

v

Contents

1 Exploring the Riemann and Definite Integral ........................... 1

1.1 Examining the Integrability of Functions ........................... 2

1.2 Finding Riemann Integrals by Definition ........................... 12

1.3 Finding Limits of Certain Sequences. . . . . ........................... 20

1.4 CalculatingVarious Interesting Definite Integrals.................. 25

1.5 Explaining Several Apparent Paradoxes . ........................... 37

1.6 Using the (Second) Mean Value Theorem . . . ....................... 45

1.7 Exercises for Independent Work .. . .................................. 49

2 Examining Improper Integrals............................................ 53

2.1 Investigating the Convergence of Integrals by Definition . . ........ 54

2.2 Using Different Criteria.............................................. 60

2.3 Using Integral Test for Convergence of Series. . . ................... 70

2.4 Exercises for Independent Work .. . .................................. 74

3 Applying One-Dimensional Integrals to Geometry and Physics ...... 75

3.1 Finding Lengths of Curves. .......................................... 76

3.2 Calculating Areas of Surfaces . ...................................... 83

3.3 Finding Volumes and Surface Areas of Solids ofRevolution...... 91

3.4 Finding Various Physical Quantities . .. . ............................ 98

3.5 Exercises for Independent Work .. . .................................. 108

4 Dealing with Functions of Several Variables ............................ 111

4.1 Finding Images and Preimages of Sets .. . ........................... 112

4.2 Examining Limits and Continuity of Functions . ................... 115

4.3 Exercises for Independent Work .. . .................................. 123

5 Investigating Derivatives of Multivariable Functions .................. 125

5.1 Calculating Partial and Directional Derivatives . . ................... 126

5.2 Examining Differentiability of Functions . .. ........................ 130

5.3 Exercises for Independent Work .. . .................................. 140

vii

viii Contents

6 Examining Higher Derivatives, Differential Expressions,

and Taylor's Formula ...................................................... 141

6.1 Verifying the Existence of the Second Derivatives . . .. . ... . ........ 142

6.2 TransformingDifferentialExpressionsand Operators ............. 147

6.3 Expanding Functions . . . . .. . .......................................... 158

6.4 Exercises for Independent Work .. . .................................. 165

7 Examining Extremes and Other Important Points ..................... 167

7.1 Looking for Global Maxima and Minima of Functions

on Compact Sets ...................................................... 168

7.2 Examining Local Extremes and Saddle Points of Functions .. . .... 175

7.3 Exercises for Independent Work .. . .................................. 186

8 Examining Implicit and Inverse Functions .............................. 187

8.1 Investigating the Existence and Extremes of Implicit Functions. . . 188

8.2 Finding Derivatives of Inverse Functions ........................... 201

8.3 Exercises for Independent Work .. . .................................. 205

9 Solving Differential Equations of the First Order ...................... 207

9.1 Finding Solutions of Separable Equations . .. ....................... 208

9.2 Solving Homogeneous Equations . .................................. 216

9.3 Solving Several Specific Equations . . .. . ............................ 221

9.4 Solving Exact Equations . . . .......................................... 234

9.5 Exercises for Independent Work .. . .................................. 249

10 Solving Differential Equations of Higher Orders ....................... 251

10.1 Solving Linear Equations with Constant Coefficients .. ............ 252

10.2 Using Various Specific Methods . . . .................................. 265

10.3 Exercises for Independent Work .. . .................................. 277

11 Solving Systems of First-Order Differential Equations ................ 279

11.1 Using the Method of Elimination of Variables...................... 280

11.2 Solving Systems of Linear Equations with Constant

Coefficients ........................................................... 286

11.3 Exercises for Independent Work .. . .................................. 310

12 Integrating in Many Dimensions .......................................... 313

12.1 Examining the Integrability of Functions . . .. ....................... 315

12.2 Calculating Integrals on Given Domains............................ 322

12.3 Changing the Order of Integration. .................................. 328

12.4 Exercises for Independent Work .. . .................................. 334

13 Applying Multidimensional Integrals to Geometry and Physics ...... 337

13.1 Finding Areas of Flat Surfaces ...................................... 338

13.2 Calculating Volumes of Solids....................................... 345

13.3 Finding Center-of-Mass Locations .................................. 354

Contents ix

13.4 Calculating Moments ofInertia...................................... 361

13.5 Finding Various Physical Quantities . .. . ............................ 368

13.6 Exercises for Independent Work .. . .................................. 378

Index ............................................................................... 381

Definitions and Notation

In this book, the conventions and definitions adopted in the first part are used.

Several additional designations are listed below.

• The symbol

[a,b]

f(x)dx denotes the Riemann integral on the interval [ a,b], and

b

a

f(x)dx refers to the definite integral.

• The iterated limits are written in the form lim

x→ alim

y→ bf(x,y)

or lim

y→ blim

x→ af(x,y), while the full (i.e., non-iterated) limit is denoted as

lim

(x,y )→ (a,b) f (x,y).

• For typographic reasons, a column vector will be written in the text as

[v1 ,v

2,...,v

N]without adding any additional transposition index.

• The symbol 1 1 is used for the unit matrix, the dimension of which varies

depending on the context.

• The symbol f

hdenotes the directional derivative along a given vector h.

• For the function of several variables, f refers to the whole Jacobian matrix, and

f to the matrix of second derivatives.

• The symbol v denotes the norm of a vector v . In the case of x∈R N ,the

notation | x|= x2

1+x 2

2+...+ x 2

Nis interchangeably used. Sometimes, the

arrow over x is omitted if it is obvious from the context that x∈R N .

xi

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