Textbook
Introduction to Integral Calculus: Systematic Studies with Engineering Applications for BeginnersJanuary 2012, ©2012

I ntegration is an important function of calculus, and Introduction to Integral Calculus combines fundamental concepts with scientific problems to develop intuition and skills for solving mathematical problems related to engineering and the physical sciences. The authors provide a solid introduction to integral calculus and feature applications of integration, solutions of differential equations, and evaluation methods. With logical organization coupled with clear, simple explanations, the authors reinforce new concepts to progressively build skills and knowledge, and numerous realworld examples as well as intriguing applications help readers to better understand the connections between the theory of calculus and practical problem solving.
The first six chapters address the prerequisites needed to understand the principles of integral calculus and explore such topics as antiderivatives, methods of converting integrals into standard form, and the concept of area. Next, the authors review numerous methods and applications of integral calculus, including:

Mastering and applying the first and second fundamental theorems of calculus to compute definite integrals

Defining the natural logarithmic function using calculus

Evaluating definite integrals

Calculating plane areas bounded by curves

Applying basic concepts of differential equations to solve ordinary differential equations
With this book as their guide, readers quickly learn to solve a broad range of current problems throughout the physical sciences and engineering that can only be solved with calculus. Examples throughout provide practical guidance, and practice problems and exercises allow for further development and finetuning of various calculus skills. Introduction to Integral Calculus is an excellent book for upperundergraduate calculus courses and is also an ideal reference for students and professionals who would like to gain a further understanding of the use of calculus to solve problems in a simplified manner.
PREFACE xiii
BIOGRAPHIES xxi
INTRODUCTION xxiii
ACKNOWLEDGMENT xxv
1 Antiderivative(s) [or Indefinite Integral(s)] 1
1.1 Introduction 1
1.2 Useful Symbols, Terms, and Phrases Frequently Needed 6
1.3 Table(s) of Derivatives and their corresponding Integrals 7
1.4 Integration of Certain Combinations of Functions 10
1.5 Comparison Between the Operations of Differentiation and Integration 15
2 Integration Using Trigonometric Identities 17
2.1 Introduction 17
2.2 Some Important Integrals Involving sin x and cos x 34
2.3 Integrals of the Form ? (d/( a sin + b cos x)), where a, b
ϵ r 37
3a Integration by Substitution: Change of Variable of Integration 43
3b Further Integration by Substitution: Additional Standard Integrals 67
4a Integration by Parts 97
4b Further Integration by Parts: Where the Given Integral Reappears on RightHand Side 117
5 Preparation for the Definite Integral: The Concept of Area 139
5.1 Introduction 139
5.2 Preparation for the Definite Integral 140
5.3 The Definite Integral as an Area 143
5.4 Definition of Area in Terms of the Definite Integral 151
5.5 Riemann Sums and the Analytical Definition of the Definite Integral 151
6a The Fundamental Theorems of Calculus 165
6b The Integral Function Ð x 1 1 t dt, (x > 0) Identified as ln x or loge x 183
7a Methods for Evaluating Definite Integrals 197
7b Some Important Properties of Definite Integrals 213
8a Applying the Definite Integral to Compute the Area of a Plane Figure 249
8b To Find Length(s) of Arc(s) of Curve(s), the Volume(s) of Solid(s) of Revolution, and the Area(s) of Surface(s) of Solid(s) of Revolution 295
9a Differential Equations: Related Concepts and Terminology 321
9a.4 Definition: Integral Curve 332
9b Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree 361
INDEX 399
G. C. JAIN, B.Sc., is a retired scientist from the Defense Research and Development Organization in India.
AJAY K. PODDAR, PhD, is Chief Scientist at Synergy Microwave Corporation. A Senior Member of the IEEE, Dr. Poddar holds several dozen patents and has published more than 180 scientific papers.
A. K. GHOSH, PhD, is Professor in the Department of Aerospace Engineering at the IIT Kanpur, India. He has published more than 120 scientific papers.
“Introduction to Integral Calculus is an excellent book for upperundergraduate calculus courses and is also an ideal reference for students and professionals who would like to gain a further understanding of the use of calculus to solve problems in a simplified manner.” (Zentralblatt MATH, 2012)
“Long on examples but often short of exercises, this work might best be used as a reference source. Summing Up: Recommended. Lowerand upperdivision undergraduates.” (Choice, 1 September 2012)
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