Introduction to Integral Calculus Systematic Studies with Engineering Applications for Beginners
by Rohde, Ulrich L.; Jain, G. c.; Poddar, Ajay K.; Ghosh, A. K.Edition: 1st
Format: Hardcover
Pub. Date: 2012-01-03
Publisher(s): Wiley
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Summary
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
| 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 logex | 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 ex] | 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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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