Integral Calculus for Engineers

The book mainly deals with basic concepts and examples about integral calculus such as indefinite integral, definite integral, improper integrals, integrals dependent on parameters, lines integrals, double and triple integrals, and surface integrals. These basic elements of integral calculus are well presented in this book, and they are indispensable for students in higher technical education to successfully approach other theoretical or technical disciplines. Preface Contents 1 Indefinite Integrals 1.1 The Notion of Primitive Function (Antiderivative) of a Function 1.2 Basic Properties of Indefinite Integrals 1.3 Primitives of Rational Functions 1.4 Primitives of Trigonometric Functions 1.5 Primitives of Irrational Functions. Binomial Integrals 2 Definite Integrals 2.1 Area of a Curvilinear Trapezoid 2.2 Darboux Sums. Definition of Definite Integral 2.3 Integrability of Continuous and Monotonic Functions 2.4 Riemann Sums. Riemann Criterion for Integrability 2.5 Lebesgue Criterion for Integrability 2.6 Properties of Integrable Functions 2.7 Area of a Plane Figure 2.8 Approximating Definite Integral 3 Improper Integrals 3.1 Convergence and Divergence of Improper Integrals 3.2 Convergence Criteria for Improper Integrals 4 Integrals Depending on Parameter 4.1 Proper Integrals Depending on a Parameter 4.2 Improper Integrals Depending on a Parameter 4.3 Euler Integrals 5 Line Integrals 5.1 Parameterized Paths. Definition of a Curve 5.2 Rectifiable Paths and Curves 5.3 Natural Parameterization of a Curve 5.4 Line Integrals of the First Kind 5.5 Line Integrals of the Second Kind 5.6 Independence on the Path of Line Integral of the Second Kind 6 Double and Triple Integrals 6.1 Double Integral. Definition and Properties 6.2 Basic Properties of the Double Integral 6.3 Reducing a Double Integral to an Iterated Single Integral 6.4 Change of Variables in Double Integral 6.5 Applications of the Double Integral in Geometry and Mechanics 6.5.1 Mass of a Lamina 6.5.2 Coordinates of the Center of Mass of a Lamina 6.5.3 Moments of Inertia of a Lamina 6.6 Riemann–Green Formula 6.7 Improper Double Integrals 6.8 Volume of a Space Figure 6.9 Triple Integrals. Definition and Basic Properties 6.10 Computing Triple Integral. Change of Variables in Triple Integral 6.11 Applications of the Triple Integral in Geometry and Mechanics 6.11.1 Volume of a Space Domain 6.11.2 Mass of a Solid 6.11.3 Coordinates of the Center of Mass of a Solid 6.11.4 Moment of Inertia of a Solid 7 Surface Integrals 7.1 Parameterized Surface Canvases. Definition of a Surface 7.2 The Area of a Smooth Surface 7.3 Surface Integral of the First Kind 7.4 Surface Integral of the Second Kind 7.5 Integral Formulas References Index