Introduction to Integral Calculus Systematic

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.

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.

Foreword	p. ix
Preface	p. xiii
Biographies	p. xxi
Introduction	p. xxiii
Acknowledgment	p. xxv
Antiderivative(s) [or Indefinite Integral(s)]	p. 1
Introduction	p. 1
Useful Symbols, Terms, and Phrases Frequently Needed	p. 6
Table(s) of Derivatives and their corresponding Integrals	p. 7
Integration of Certain Combinations of Functions	p. 10
Comparison Between the Operations of Differentiation and Integration	p. 15
Integration Using Trigonometric Identities	p. 17
Introduction	p. 17
Some Important Integrals Involving sin x and cos x	p. 34
Integrals of the Form (dx/(a sin x + b cos x)), where a, b r	p. 37
Integration by Substitution: Change of Variable of Integration	p. 43
Introduction	p. 43
Generalized Power Rule	p. 43
Theorem	p. 46
To Evaluate Integrals of the Form , where a, b, c, and d are constant	p. 60
Further Integration by Substitution: Additional Standard Integrals	p. 67
Introduction	p. 67
Special Cases of Integrals and Proof for Standard Integrals	p. 68
Some New Integrals	p. 84
Four More Standard Integrals	p. 85
Integration by Parts	p. 97
Introduction	p. 97
Obtaining the Rule for Integration by Parts	p. 98
Helpful Pictures Connecting Inverse Trigonometric Functions with Ordinary Trigonometric Functions	p. 113
Rule for Proper Choice of First Function	p. 115
Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Side	p. 117
Introduction	p. 117
An Important Result: A Corollary to Integration by Parts	p. 120
Application of the Corollary to Integration by Parts to Integrals that cannot be Solved Otherwise	p. 124
Simpler Method(s) for Evaluating Standard Integrals	p. 126
To Evaluate	p. 133
Preparation for the Definite Integral: The Concept of Area	p. 139
Introduction	p. 139
Preparation for the Definite Integral	p. 140
The Definite Integral as an Area	p. 143
Definition of Area in Terms of the Definite Integral	p. 151
Riemann Sums and the Analytical Definition of the Definite Integral	p. 151
The Fundamental Theorems of Calculus	p. 165
Introduction	p. 165
Definite Integrals	p. 165
The Area of Function A(x)	p. 167
Statement and Proof of the Second Fundamental Theorem of Calculus	p. 171
Differentiating a Definite Integral with Respect to a Variable Upper Limit	p. 172
The Integral Function Identified as lnx or log_ex	p. 183
Introduction	p. 183
Definition of Natural Logarithmic Function	p. 186
The Calculus of lnx	p. 187
The Graph of the Natural Logarithmic Function lnx	p. 194
The Natural Exponential Function [exp(x) or e^x]	p. 196
Methods for Evaluating Definite Integrals	p. 197
Introduction	p. 197
The Rule for Evaluating Definite Integrals	p. 198
Some Rules (Theorems) for Evaluation of Definite Integrals	p. 200
Method of Integration by Parts in Definite Integrals	p. 209
Some Important Properties of Definite Integrals	p. 213
Introduction	p. 213
Some Important Properties of Definite Integrals	p. 213
Proof of Property (P₀)	p. 214
Proof of Property (P₅)	p. 228
Definite Integrals: Types of Functions	p. 232
Applying the Definite Integral to Compute the Area of a Plane Figure	p. 249
Introduction	p. 249
Computing the Area of a Plane Region	p. 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 Revolution	p. 295
Introduction	p. 295
Methods of Integration	p. 295
Equation for the Length of a Curve in Polar Coordinates	p. 300
Solids of Revolution	p. 302
Formula for the Volume of a "Solid of Revolution"	p. 303
Area(s) of Surface(s) of Revolution	p. 314
Differential Equations: Related Concepts and Terminology	p. 321
Introduction	p. 321
Important Formal Applications of Differentials (dy and dx)	p. 323
Independent Arbitrary Constants (or Essential Arbitrary Constants)	p. 331
Definition: Integral Curve	p. 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 Equations	p. 357
Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree	p. 361
Introduction	p. 361
Methods of Solving Differential Equations	p. 362
Linear Differential Equations	p. 388
Type III: Exact Differential Equations	p. 397
Applications of Differential Equations	p. 398
INDEX	p. 399
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Introduction to Integral Calculus Systematic Studies with Engineering Applications for Beginners

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