Introduction to Integral Calculus by Rohde and GC Jain
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Introduction to Integral Calculus (Systematic Studies with Engineering Applications for Beginners) written by
Ulrich L. Rohde, Prof. Dr.-Ing. Dr. h. c. mult., BTU Cottbus, Germany, Synergy Microwave Corporation Peterson, NJ, USA
G. C. Jain (Retd. Scientist) Defense Research and Development Organization, Maharashtra, India
Ajay K. Poddar, Chief Scientist, Synergy Microwave Corporation, Peterson, NJ, USA
A. K. Ghosh, Professor, Department of Aerospace Engineering, Indian Institute of Technology – Kanpur, Kanpur, India
The purpose of these works is to provide the basic (but solid) foundation of Calculus to beginners. The books aim to show them the enjoyment in the beauty and power of Calculus and develop the ability to select proper material needed for their studies in any technical and scientific field, involving Calculus
The author’s aim throughout has been to provide a tour of Calculus for a beginner as well as strong fundamental basics to undergraduate students on the basis of the following questions, which frequently came to our minds, and for which we wanted satisfactory and correct answers.
(i) What is Calculus?
(ii) What does it calculate?
(iii) Why do teachers of physics and mathematics frequently advise us to learn Calculus seriously?
(iv) How is Calculus more important and more useful than algebra and trigonometry or any other branch of mathematics?
(v) Why is Calculus more difficult to absorb than algebra or trigonometry?
(vi) Are there any problems faced in our day-to-day life that can be solved more easily by Calculus than by arithmetic or algebra?
(vii) Are there any problems which cannot be solved without Calculus?
(viii) Why study Calculus at all?
(ix) Is Calculus different from other branches of mathematics?
(x) What type(s) of problems are handled by Calculus?
Introduction to Integral Calculus (Systematic Studies with Engineering Applications for Beginners) written by Rohde and GC Jain
cover the following topics.
FOREWORD
PREFACE
BIOGRAPHIES
INTRODUCTION
ACKNOWLEDGMENT
1. Antiderivative(s) [or Indefinite Integral(s)]
1.1 Introduction
1.2 Useful Symbols, Terms, and Phrases Frequently Needed
1.3 Table(s) of Derivatives and their corresponding Integrals
1.4 Integration of Certain Combinations of Functions
1.5 Comparison Between the Operations of Differentiation and Integration
2. Integration Using Trigonometric Identities
2.1 Introduction
2.2 Some Important Integrals Involving sin x and cos x
2.3 Integrals of the Form Ððdx=ða sin x þ b cos xÞÞ, where a, b 2 r
3a. Integration by Substitution: Change of Variable of Integration
3a.1 Introduction
3a.2 Generalized Power Rule
3a.3 Theorem
3a.4 To Evaluate Integrals of the Form ð a sin x þ b cos x c sin x þ d cos x dx; where a, b, c, and d are constant
3b. Further Integration by Substitution: Additional Standard Integrals
3b.1 Introduction
3b.2 Special Cases of Integrals and Proof for Standard Integrals
3b.3 Some New Integrals
3b.4 Four More Standard Integrals
4a. Integration by Parts
4a.1 Introduction
4a.2 Obtaining the Rule for Integration by Parts
4a.3 Helpful Pictures Connecting Inverse Trigonometric Functions with Ordinary Trigonometric Functions
4a.4 Rule for Proper Choice of First Function
4b. Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Side
4b.1 Introduction
4b.2 An Important Result: A Corollary to Integration by Parts
4b.3 Application of the Corollary to Integration by Parts to Integrals that cannot be Solved Otherwise
4b.4 Simpler Method(s) for Evaluating Standard Integrals
4b.5 To Evaluate
5. Preparation for the Definite Integral: The Concept of Area
5.1 Introduction
5.2 Preparation for the Definite Integral
5.3 The Definite Integral as an Area
5.4 Definition of Area in Terms of the Definite Integral
5.5 Riemann Sums and the Analytical Definition of the Definite Integral
6a. The Fundamental Theorems of Calculus
6a.1 Introduction
6a.2 Definite Integrals
6a.3 The Area of Function A(x)
6a.4 Statement and Proof of the Second Fundamental Theorem of Calculus
6a.5 Differentiating a Definite Integral with Respect to a Variable Upper Limit
6b. The Integral Function Ð x11t dt, (x > 0) Identified as ln x or loge x
6b.1 Introduction
6b.2 Definition of Natural Logarithmic Function
6b.3 The Calculus of ln x
6b.4 The Graph of the Natural Logarithmic Function ln x
6b.5 The Natural Exponential Function [exp(x) or ex]
7a. Methods for Evaluating Definite Integrals
7a.1 Introduction 197
7a.2 The Rule for Evaluating Definite Integrals
7a.3 Some Rules (Theorems) for Evaluation of Definite Integrals
7a.4 Method of Integration by Parts in Definite Integrals
7b Some Important Properties of Definite Integrals
7b.1 Introduction
7b.2 Some Important Properties of Definite Integrals
7b.3 Proof of Property (P0)
7b.4 Proof of Property (P5)
7b.5 Definite Integrals: Types of Functions
8a. Applying the Definite Integral to Compute the Area of a Plane Figure
8a.1 Introduction
8a.2 Computing the Area of a Plane Region
8a.3 Constructing the Rough Sketch [Cartesian Curves]
8a.4 Computing the Area of a Circle (Developing Simpler Techniques)
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
8b.1 Introduction
8b.2 Methods of Integration
8b.3 Equation for the Length of a Curve in Polar Coordinates
8b.4 Solids of Revolution
8b.5 Formula for the Volume of a “Solid of Revolution”
8b.6 Area(s) of Surface(s) of Revolution
9a. Differential Equations: Related Concepts and Terminology
9a.1 Introduction
9a.2 Important Formal Applications of Differentials (dy and dx)
9a.3 Independent Arbitrary Constants (or Essential Arbitrary Constants)
9a.4 Definition: Integral Curve
9a.5 Formation of a Differential Equation from a Given Relation, Involving Variables and the Essential Arbitrary Constants (or Parameters)
9a.6 General Procedure for Eliminating “Two” Independent Arbitrary Constants (Using the Concept of Determinant)
9a.7 The Simplest Type of Differential Equations
9b. Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree
9b.1 Introduction
9b.2 Methods of Solving Differential Equations
9b.3 Linear Differential Equations
9b.4 Type III: Exact Differential Equations
9b.5 Applications of Differential Equations
INDEX
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