Introduction to Integral Calculus : Systematic Studies with Engineering Applications for Beginners.
Title:
Introduction to Integral Calculus : Systematic Studies with Engineering Applications for Beginners.
Author:
Rohde, Ulrich L.
ISBN:
9781118130315
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (430 pages)
Contents:
Introduction to Integral Calculus: Systematic Studies with Engineering Applications for Beginners -- CONTENTS -- 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 ∫(dx/(asin x + b cosx)), where a, Є 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 ∫ a sin x + b cos x/c sin x + 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 ∫√ax2 + bx + cdx.
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 ∫x1 1/ t dt, (x > 0) Identified as ln x or logex -- 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 -- 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.
Abstract:
An accessible introduction to the fundamentals of calculus needed to solve current problems in engineering and the physical sciences 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 real-world 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 anti-derivatives, 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 fine-tuning of various calculus skills.
Introduction to Integral Calculus is an excellent book for upper-undergraduate 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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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