Integrals Vol 2

199 Pages·2018·2.371 MB·English

by  Demetrios P. Kanoussis, Kalamos Attikis

[1]

Integrals Vol. 2
The Definite Integral

1) An excellent supplementary text for all Mathematics, Engineering and
Technology students, ideal for independent study
2) 130 fully solved illustrative examples and 260 problems for solution
3) Evaluation techniques and methods and various applications
4) Odd numbered problems are provided with answers
5) Hints or detailed outlines are given for the more involved problems
[2]

About the Author
Demetrios P. Kanoussis, Ph.D
Kalamos Attikis, Greece
[email protected]

Dr. Kanoussis is a professional Electrical Engineer and Mathematician. He
received his Ph.D degree in Engineering and his Master degree in Mathematics
from Tennessee Technological University, U.S.A, and his Bachelor degree in
Electrical Engineering from the National Technical University of Athens
(N.T.U.A), Greece.
As a professional Electrical Engineer, Dr. Kanoussis has been actively involved
in the design and in the implementation of various projects, mainly in the area
of the Integrated Control Systems.
Regarding his teaching experience, Dr. Kanoussis has long teaching experience
in the field of Applied Mathematics and Electrical Engineering.
His original scientific research and contribution, in Mathematics and Electrical
Engineering, is published in various, high impact international journals.
Additionally to his professional activities, teaching and research, Demetrios P.
Kanoussis is the author of several textbooks in Electrical Engineering and
Applied Mathematics.
[3]

Integrals Vol. 2
The Definite Integral
Copyright 2018, Author: Demetrios P. Kanoussis.
All rights reserved. No part of this publication may be reproduced, distributed
or transmitted in any form or by any means, electronic or mechanical, without
the prior written permission of the author, except in the case of brief
quotations and certain other noncommercial uses permitted by copyright law.
Inquires should be addressed directly to the author,
Demetrios P. Kanoussis
[email protected]
This e book is licensed for your personal use only. This e book may not be
resold or given away to other people. If you would like to share this book with
another person, please purchase an additional copy for each recipient.
Thank you for respecting the work of this author.
First edition: May 2018.
[4]

PREFACE

In solving various problems in Engineering, Physics and Geometry we have to
sum up an infinite number of infinitesimal quantities (summands). This leads
to the notion of the Definite Integral which is one of the most important
concepts in Mathematics.
Archimedes (287-211 BC) the great Greek Mathematician and Engineer of
antiquity, using his famous “method of exhaustion” was able to evaluate areas
of curvilinear plane figures. This method is considered to be the precursor of
the contemporary Integral Calculus, discovered independently by Newton
(1642-1726) and Leibniz (1646-1716) in the mid-17th century.
Indefinite Integrals are studied in considerable depth and extent in my e book
“Integrals, Vol. 1, The Indefinite Integral”. In this volume we study the
“Definite Integral” which is connected to the Indefinite Integral by the so
called “The fundamental Theorem of Integral Calculus, (The Newton-Leibniz
Theorem)”.
This book is applications oriented and has been designed to be an excellent
supplementary book for University and College students in all areas of
Mathematics, Physics and Engineering.
The content of the book is divided into 20 chapters as shown analytically in the
Table of Contents.
In the first five chapters we consider some examples leading directly to the
“heart” of the notion of the Definite Integral and study some fundamental
properties of the integrals, i.e. integrating finite sums of functions, integrating
inequalities, The Mean Value Theorem of Integral Calculus, etc.
In chapter 6 we state and prove the two Fundamental Theorems of Integral
Calculus.
In chapter 7 we develop methods of evaluating Definite Integrals with the aid
of the corresponding Indefinite Integrals or by the powerful method of
substitution.
In chapter 8 we study the integration of complex functions of real arguments.
[5]

In chapter 9 we define the mean or average value of a function over some
finite interval and derive the fundamental formula for the mean value in terms
of a definite integral.
Chapters 10 and 11 are devoted to the estimation of sums by definite
integrals and the definite integrals of even, odd and periodic functions.
In chapter 12 we consider the problem of evaluating areas bounded by plane
figures (defined in Cartesian or Polar coordinates or in parametric form) with
the aid of Definite Integrals.
In chapter 13 we evaluate the length of arcs of curves expressed either in
Cartesian or Polar coordinates.
In chapter 14 we study the computation of volumes of solids.
In chapter 15 we evaluate the area of a surface of revolution.
In chapter 16 we study the center of gravity of various plane or solid figures
for either a discrete or a continuous mass distribution.
In chapter 17 we state and prove the two Theorems of Pappus of Alexandria
and consider various applications.
In chapter 18 we consider the numerical (approximate) integration, i.e. the
Trapezoidal formula, the Simpson’s rule, integration by expanding the
integrant into a power series, the Gauss’s quadrature, etc.
In chapter 19 we study the so called “Improper Integrals” which appear quite
naturally in various applications. The “Cauchy Principal Value of an improper
integral” is defined and various applications are considered.
In chapter 20 we consider applications of the Definite Integral in Physics and
Engineering, (work of a variable force, distance and displacement, pressure
force, power and energy in electric circuits, etc).
The text includes 130 illustrative worked out examples and 260 graded
problems to be solved. The examples and the problems are designed to help
the students to develop a solid background in the evaluation of Integrals, to
broaden their knowledge and sharpen their analytical skills and finally to
[6]

prepare them to pursue successful studies in more advanced courses in
Mathematics.
A brief hint or a detailed outline in solving more involved problems is often
given.
[7]

Table of Contents
CHAPTER 1 Some simple examples leading to the concept of the 9
Definite Integrals

CHAPTER 2 The definition of the Definite Integral 20

CHAPTER 3 Fundamental properties of Definite Integrals 28

CHAPTER 4 Integrating inequalities 35

CHAPTER 5 The mean value theorem of Integral Calculus 41

CHAPTER 6 The two fundamental theorems of Integral Calculus 46

CHAPTER 7 Methods of evaluating Definite Integrals 57

CHAPTER 8 Integrating complex functions of real arguments 68

CHAPTER 9 The mean value of a function 71

CHAPTER 10 Estimates of sums by Definite Integrals 75

CHAPTER 11 Definite Integrals of even and odd functions 77

CHAPTER 12 Areas of plane figures 84

CHAPTER 13 The arc length of a curve 102

CHAPTER 14 Volumes of solids 113

CHAPTER 15 The area of a surface of revolution 123

CHAPTER 16 Centers of gravity 132

CHAPTER 17 The two theorems of Pappus of Alexandria 146

CHAPTER 18 Numerical integration 149
[8]

CHAPTER 19 Improper Integrals (a brief introduction) 159

CHAPTER 20 Applications of Definite Integrals in Engineering and 176
Physics