Introduction to Integral Calculus : Systematic Studies with Engineering Applications for Beginners

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  • Format: Hardcover
  • Copyright: 2012-01-03
  • Publisher: Wiley

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This book explores the integral calculus and its plentiful applications in engineering and the physical sciences. The authors aim to develop a basic understanding of integral calculus combined with scientific problems, and throughout, the book details the numerous applications of calculus as well as presents the topic as a deep, rich, intellectual achievement. The needed fundamental information is presented in addition to plentiful references, exercises, and examples. The definition of an integral is motivated by the familiar notion of area. Although the methods of plane geometry allow for the areas of polygons to be calculated, they do not provide ways of finding the area of plane regions whose boundaries are curves other than circles. By means of the integral, the areas of many such regions can be found. The authors also use this definition to calculate volumes and length of curves etc. Topical coverage includes anti-differentiation; integration of trigonometric functions; integration by substitution; methods of substitution; the definite integral; methods for evaluating definite integrals; differential equations and their solutions; and ordinary differential equations of first order and first degree.

Author Biography

Ulrich L. Rohde, PhD, ScD, Dr-Ing, is Chairman of Synergy Microwave Corporation, President of Communications Consulting Corporation, and a Partner of Rohde Schwarz. A Fellow of the IEEE, Professor Rohde holds several patents and has published more than 200 scientific papers. G. C. Jain, BSc., 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.

Table of Contents

Forewordp. ix
Prefacep. xiii
Biographiesp. xxi
Introductionp. xxiii
Acknowledgmentp. xxv
Antiderivative(s) [or Indefinite Integral(s)]p. 1
Introductionp. 1
Useful Symbols, Terms, and Phrases Frequently Neededp. 6
Table(s) of Derivatives and their corresponding Integralsp. 7
Integration of Certain Combinations of Functionsp. 10
Comparison Between the Operations of Differentiation and Integrationp. 15
Integration Using Trigonometric Identitiesp. 17
Introductionp. 17
Some Important Integrals Involving sin x and cos xp. 34
Integrals of the Form (dx/(a sin x + b cos x)), where a, b rp. 37
Integration by Substitution: Change of Variable of Integrationp. 43
Introductionp. 43
Generalized Power Rulep. 43
Theoremp. 46
To Evaluate Integrals of the Form , where a, b, c, and d are constantp. 60
Further Integration by Substitution: Additional Standard Integralsp. 67
Introductionp. 67
Special Cases of Integrals and Proof for Standard Integralsp. 68
Some New Integralsp. 84
Four More Standard Integralsp. 85
Integration by Partsp. 97
Introductionp. 97
Obtaining the Rule for Integration by Partsp. 98
Helpful Pictures Connecting Inverse Trigonometric Functions with Ordinary Trigonometric Functionsp. 113
Rule for Proper Choice of First Functionp. 115
Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Sidep. 117
Introductionp. 117
An Important Result: A Corollary to Integration by Partsp. 120
Application of the Corollary to Integration by Parts to Integrals that cannot be Solved Otherwisep. 124
Simpler Method(s) for Evaluating Standard Integralsp. 126
To Evaluatep. 133
Preparation for the Definite Integral: The Concept of Areap. 139
Introductionp. 139
Preparation for the Definite Integralp. 140
The Definite Integral as an Areap. 143
Definition of Area in Terms of the Definite Integralp. 151
Riemann Sums and the Analytical Definition of the Definite Integralp. 151
The Fundamental Theorems of Calculusp. 165
Introductionp. 165
Definite Integralsp. 165
The Area of Function A(x)p. 167
Statement and Proof of the Second Fundamental Theorem of Calculusp. 171
Differentiating a Definite Integral with Respect to a Variable Upper Limitp. 172
The Integral Function Identified as lnx or logexp. 183
Introductionp. 183
Definition of Natural Logarithmic Functionp. 186
The Calculus of lnxp. 187
The Graph of the Natural Logarithmic Function lnxp. 194
The Natural Exponential Function [exp(x) or ex]p. 196
Methods for Evaluating Definite Integralsp. 197
Introductionp. 197
The Rule for Evaluating Definite Integralsp. 198
Some Rules (Theorems) for Evaluation of Definite Integralsp. 200
Method of Integration by Parts in Definite Integralsp. 209
Some Important Properties of Definite Integralsp. 213
Introductionp. 213
Some Important Properties of Definite Integralsp. 213
Proof of Property (P0)p. 214
Proof of Property (P5)p. 228
Definite Integrals: Types of Functionsp. 232
Applying the Definite Integral to Compute the Area of a Plane Figurep. 249
Introductionp. 249
Computing the Area of a Plane Regionp. 252
Constructing the Rough Sketch [Cartesian Curves]p. 257
Computing the Area of a Circle (Developing Simpler Techniques)p. 272
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 Revolutionp. 295
Introductionp. 295
Methods of Integrationp. 295
Equation for the Length of a Curve in Polar Coordinatesp. 300
Solids of Revolutionp. 302
Formula for the Volume of a "Solid of Revolution"p. 303
Area(s) of Surface(s) of Revolutionp. 314
Differential Equations: Related Concepts and Terminologyp. 321
Introductionp. 321
Important Formal Applications of Differentials (dy and dx)p. 323
Independent Arbitrary Constants (or Essential Arbitrary Constants)p. 331
Definition: Integral Curvep. 332
Formation of a Differential Equation from a Given Relation, Involving Variables and the Essential Arbitrary Constants (or Parameters)p. 333
General Procedure for Eliminating "Two" Independent Arbitrary Constants (Using the Concept of Determinant)p. 338
The Simplest Type of Differential Equationsp. 357
Methods of Solving Ordinary Differential Equations of the First Order and of the First Degreep. 361
Introductionp. 361
Methods of Solving Differential Equationsp. 362
Linear Differential Equationsp. 388
Type III: Exact Differential Equationsp. 397
Applications of Differential Equationsp. 398
INDEXp. 399
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