MATHEMATICAL ANALYSIS II INTEGRAL CALCULUS

407 Pages·2012·3.09 MB·English

MATHEMATICAL ANALYSIS II
INTEGRAL CALCULUS
SEVER ANGEL POPESCU
To my family.
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To those who really try to put their lives in the Light of
Truth.
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Dedicated to the memory of my late Professor and
Mentor, Dr. Doc. Nicolae Popescu.
Contents
Preface 7
Chapter 1. Indefinite integrals (Primitives, Antiderivatives) 1
1. Definitions, some properties and basic formulas 1
2. Some results on polynomials 10
3. Primitives of rational functions 21
4. Primitives of irrational and trigonometric functions 27
5. Problems and exercises 47
Chapter 2. Definite integrals 49
1. The mass of a linear bar 49
2. Darboux sums and their applications 52
3. Lebesgue criterion and its applications 59
4. Mean theorem. Newton-Leibniz formula 63
5. The measure of a figure in Rn 68
6. Areas of plane figures bounded by graphics of functions 75
7. The volume of a rotational solid 85
8. The length of a curve in R3 87
9. Approximate computation of definite integrals. 90
10. Problems and exercises 101
Chapter 3. Improper (generalized) integrals 105
1. More on limits of functions of one variable 105
2. Improper integrals of the first type 112
3. Improper integrals of the second type 122
4. Problems and exercises 135
Chapter 4. Integrals with parameters 137
1. Proper integrals with parameters 137
2. Improper integrals with parameters 147
3. Euler’s functions gamma and beta 167
4. Problems and exercises 179
Chapter 5. Line integrals 181
1. The mass of a wire. Line integrals of the first type. 181
2. Line integrals of the second type. 194
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4 CONTENTS
3. Independence on path. Conservative fields 201
4. Computing plane areas with line integrals 211
5. Supplementary remarks on line integrals 216
6. Problems and exercises 221
Chapter 6. Double integrals 225
1. Double integrals on rectangles. 225
2. Double integrals on an arbitrary bounded domain 233
3. Green formula. Applications. 244
4. Change of variables in double integrals. Polar coordinates. 251
5. Problems and exercises 263
Chapter 7. Triple integrals 265
1. What is a triple integral on a parallelepiped? 265
2. Triple integrals on a general domain. 271
3. Iterative formulas for a general space domain 278
4. Change of variables in a triple integral 286
5. Problems and exercises 303
Chapter 8. Surface integrals 305
1. Deformation of a 2D-domain into a space surface 305
2. Surface integral of the first type. The mass of a 3D-lamina.313
3. Flux of a vector field through an oriented surface. 319
4. Problems and exercises 335
Chapter 9. Gauss and Stokes formulas. Applications 339
1. Gauss formula (divergence theorem) 339
2. A mathematical model of the heat flow in a solid 348
3. Stokes Theorem 349
4. Problems and exercises 357
Chapter 10. Some remarks on complex functions integration 359
1. Complex functions integration 359
2. Applications of residues formula 377
3. Problems and exercises 385
Appendix A. Exam samples 387
1. June, 2010 387
2. June, 2010 387
3. June, 2010 387
4. June, 2010 388
5. September, 2010 388
6. September, 2010 388
Appendix B. Basic antiderivatives 389
CONTENTS 5
Appendix. Index 391
Appendix. Bibliography 397

Preface
Here is the second volume "Mathematical Analysis II. Integral Cal-
culus" of my course of Advanced Calculus for Engineers and beginning
Mathematicians. The first volume "Mathematical Analysis I. Differ-
ential Calculus" appeared in 2009 (see [Po]). I invite the reader to
carefully meditate on the main ideas I discussed in the Preface of this
firstvolume. ThematterisaboutthewayIthinkMathematicalAnaly-
sis(orAdvancedCalculus)havetobetaughttoanengineeringstudent.
First of all I insist on a strong and correct intuitive basis of any math-
ematical notion and statement. Then I proceed step by step to a rig-
orously mathematical model of this notion or statement. My aim in
this course (primarily intended forengineering students) is to create an
intuitive thinking to the reader who must feel the simplicity and the
naturalness of the introduced notion or concept. Abstract Mathemat-
ics need not destroy the intuition of the future engineer. Mathematical
courses must make this intuition stronger, more mature and very solid.
This is why this course appeared to be "a spoken course". I wanted it
to be so! In general, I like a lot "live" Mathematics. A "dead" abstract
notion, without at least one small motivation, generally will remain
dead and nonuseful for an engineering student.
I also blame the opposite side, that mathematical teaching which
consists only on a sequence of examples, motivations, etc., without a
general definition, without a mathematical correct reasoning, based on
"stories", etc. A creative middle way teaching is to be preferred. All of
these belong to something which is beyond Mathematics itself, namely
to the "Art of Teaching".
Forinstance, manytimesinthisvolume, theideaoftheconvergence
of a system A of numbers to a number I, was substituted with the idea
of a "well approximation" of I with numbers of A. Surely, I could use
from the very beginning the definition "with ε”and everything could
appeared formally rigorous for an "extremist" mathematician, but this
course is not intended for such a people! If I had done so, I would
have killed the intuitive power of the idea of approximation, extremely
useful for a future engineer and not only for him!
7
8 PREFACE
In fact, in all my courses of Mathematical Analysis, I carried on the
basic idea of Prof. Dr. Gavril Pa˘ltineanu to make appropriate mathe-
matical courses for engineers and not for mathematicians. I think that
even a student in abstract Mathematics could find a good opportunity
to read such an "applicable" course.
This book contains ten chapters and an appendix with exam sam-
ples and a table with some basic formulas for primitives (antideriva-
tives). Since I successively introduced in the first volume ([Po]) basic
elements of complex functions theory, I continued here with some ba-
sic remarks on integrals of functions with a complex variable, residues
theory and some of their applications.
I started with a serious review of the calculus of primitives, usually
assumed to be known from high school by the Romanian student.
I go on with definite or Riemann integrals, improper integrals, in-
tegrals with parameters, line integrals, double and triple integrals, sur-
face integrals, Gauss and Stokes formulas and I finish with complex
functions integrals.
Any chapter contains many motivation-examples, worked exercises
andproposedexercises. Iinsistedonusinginsteadofanabstractmath-
ematical notion more intuitive objects taken from Physics, Mechanics,
Strains of Materials, etc. Don’t say that the word "deformation" is not
better for an engineering student then the corresponding mathemati-
cal notion of "vector transformation", "parametric path", "parametric
sheet", etc.
Many people contributed directly or indirectly to this volume.
The late Professor Dr. Doc. Nicolae Popescu, my Professor and
my Master, was teaching me to do "live" Mathematics.
MycolleagueandmyfriendProf. Dr. GavriilPa˘ltineanuchallenged
me to investigate the interesting way to teach Analysis for engineers.
He gave me many deep advises on this fascinating domain which is
Mathematical Analysis. He also carried this difficult task to read this
huge volume and to push me in making many corrections.
Prof. Dr. Octav Olteanu and Prof. Dr. Nicolae Da˘ne¸t read care-
fully the entire manuscript and made some very useful suggestions.
I am also grateful to my younger colleagues Dr. Emil Popescu, Dr.
Viorel Petrehu¸s, Dr. Marilena Jianu and Dr. Valentin Popescu for
helping me a lot with the proofreading.
At last, but not the least, I will always express my thanks to two
persons: to my wife Angela for encouraging me to "go on" with this
difficult task and to my 11 years old grand-daughter Monica Izabela
2
who succeeded to refresh a lot my tired mind before chapter 4. This
PREFACE 9
is why the reader must be grateful to her for the easy (but not short)
way I succeeded to present the improper integrals with parameters.
Finally, I mention the special help that trees and birds of my close
park gave me during my silent walk around the lake in those hot
evenings of August 2010. All of these are of a great importance when
one is thinking of mathematical affairs.
Prof. Dr. Sever Angel Popescu
Technical University of Civil Engineering Bucharest
Department of Mathematics and Computer Science
B=dul Lacul Tei 124, Sector 2, Bucharest, ROMANIA
[email protected]
February, 2011