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In solving various problems in Engineering, Physics and Geometry we have to sum up an infinite number of infinitesimal quantities (summands). This leads to the notion of the Definite Integral which is one of the most important concepts in Mathematics.
Archimedes (287-211 BC) the great Greek Mathematician and Engineer of antiquity, using his famous "method of exhaustion" was able to evaluate areas of curvilinear plane figures. This method is considered to be the precursor of the contemporary Integral Calculus, discovered independently by Newton (1642-1726) and Leibniz (1646-1716) in the mid-17th century.
Indefinite Integrals are studied in considerable depth and extent in my e book "Integrals, Vol. 1, The Indefinite Integral". In this volume we study the "Definite Integral" which is connected to the Indefinite Integral by the so called "The fundamental Theorem of Integral Calculus, (The Newton-Leibniz Theorem)"
This book is applications oriented and has been designed to be an excellent supplementary book for University and College students in all areas of Mathematics, Physics and Engineering.
The content of the book is divided into 20 chapters as shown analytically
in the Table of Contents
In the first five chapters we consider some examples leading directly to the "heart" of the notion of the Definite Integral and study some fundamental properties of the integrals, i.e. integrating finite sums of functions, integrating inequalities, The Mean Value Theorem of Integral Calculus, etc.
In chapter 6 we state and prove the two Fundamental Theorems of Integral Calculus.
In chapter 7 we develop methods of evaluating Definite Integrals with the aid of the corresponding Indefinite Integrals or by the powerful method of substitution.
In chapter 8 we study the integration of complex functions of real arguments.
In chapter 9 we define the mean or average value of a function over some finite interval and derive the fundamental formula for the mean value in terms of a definite integral.
Chapters 10 and 11 are devoted to the estimation of sums by definite integrals and the definite integrals of even, odd and periodic functions.
In chapter 12 we consider the problem of evaluating areas bounded by plane figures (defined in Cartesian or Polar coordinates or in parametric form) with the aid of Definite Integrals.
In chapter 13 we evaluate the length of arcs of curves expressed either in Cartesian or Polar coordinates.
In chapter 14 we study the computation of volumes of solids.
In chapter 15 we evaluate the area of a surface of revolution.
In chapter 16 we study the center of gravity of various plane or solid figures for either a discrete or a continuous mass distribution.
In chapter 17 we state and prove the two Theorems of the Pappus of Alexandria and consider various applications.
In chapter 18 we consider the numerical (approximate) integration, i.e. the Trapezoidal formula, the Simpson's rule, integration by expanding the integrand into a power series, the Gauss's quadrature, etc.
In chapter 19 we study the so called "Improper Integrals" which appear quite naturally in various applications. The "Cauchy Principal Value of an improper integral" is defined and various applications are considered.
In chapter 20 we consider applications of the Definite Integral in Physics and Engineering, (work of a variable force, distance and displacement, pressure force, power and energy in electric circuits, etc).
The text includes 130 illustrative worked out examples and 260 graded problems to be solved. The examples and the problems are designed to help the students to develop a solid background in the evaluation of Integrals, to broaden their knowledge and sharpen their analytical skills and finally to prepare them to pursue successful studies in more advanced courses in Mathematics.
A brief hint or a detailed outline in solving more involved problems is often given.
Some simple examples leading to the concept of the
Definite Integrals
The definition of the Definite Integral
Fundamental properties of Definite Integrals
Integrating inequalities
The mean value theorem of Integral Calculus
The two fundamental theorems of Integral Calculus
Methods of evaluating Definite Integrals
Integrating complex functions of real arguments
The mean value of a function
Estimates of sums by Definite Integrals
Definite Integrals of even and odd functions Areas of plane figures
The arc length of a curve
Volumes of solids
The area of a surface of revolution
Centers of gravity
The two theorems of Pappus of Alexandria
Numerical integration
Improper Integrals (a brief introduction)
Applications of Definite Integrals in Engineering and Physics