Differential and Integral Calculus

Differential and Integral Calculus

Volume 2

4.1 ORDINARY INTEGRALS AS FUNCTIONS OF A PARAMETER	4.5. IMPROPER INTEGRALS	A4.2 GENERAL FORMULA FOB THE AREA (OR VOLUME) OF A REGION BOUNDED BY SEGMENTS OF STRAIGHT LINES OR PLANE AREAS (GULDIN'S FORMULA). THE POLAR PLANIMETER
4.1.1 Examples and Definitions	4.5.1 Functions with Jump Discontinuities	Exerxise 4.6
4.1.2. Continuity and Differentiability of an Integral with respect to the Parameter	4.5.2. Functions with Isolated Infinite Discontinuities	A4.3 VOLUMES AND AREAS IN SPACE OF ANY NUMBER OF DIMENSIONS
Exercises 4.1	4.5.3 Functions with Lines of Infinite Discontinuity	A4.3.1 Resolution of Multiple Integrals
4.2 THE INTEGRAL OF A CONTINUOUS FUNCTION OVER A REGION OF THE PLANE OR OF SPACE	4.5.4 Infinite Regions of Integration	A4.3.2 Areas of Surfaces and Integration over Surfaces in more than Three Dimensions
4.2.1 The Double Integral (Domain Integral) as a Volume	4.5.5. Summary and Extensions	A4.3.3 Area and Volume of the n-Dimensional Unit Sphere
4.2.2 The General Analytical Concept of the Integral	4.6 GEOMETRICAL APPLICATIONS	A4.3.4 Generalizations. Parametric Representations
4.2.3. Examples	4.6.1 Elementary Calculation of Volumes	Exercises 4.6
4.2.4. Notation. Extensions. Fundamental Rules	4.6.2 General Remarks on the Calculation of Volumes. Solids of Revolution. Volumes in Polar Co-ordinates	A4.4 IMPROPER INTEGRALS AS FUNCTIONS OF A PARAMETER
4.2.5 Integral Estimates and the Mean Value Theorem	4.6.3 Area of a Curved Surface	A4.4.1 Uniform Convergence. Continuous Dependence on the Parameter
4.2.6. Integrals over Regions in Three and More Dimensions	Exercises 4.4	A4.4.2 Integration and Differentiation of Improper Integrals with respect to a Parameter
4.2.7 Space Differentiation. Mass and Density	4.7 PHYSICAL APPLICATIONS	A4.4.3 Examples
4.3 REDUCTION OF THE MULTIPLE INTEGRAL TO REPEATED SINGLE INTERGRALS	4.7.1 Moments and Centre of Mass	A4.4.4 Evaluation of Fresnel Integrals:
4.3.1 Integrals over a Rectangle	4.7.2 Moment of Inertia	Exercises 4.7
4.3.2 Results. Change of Order of Integration. Differentiation under the Integral Sign	4.7.3 The Compound Pendulum	A4.5 THE FOURIER INTEGRAL
4.3.3 Extension of the Result to More General Regions	4.7.4 Potential of Attracting Masses	A4.5.1 Introduction, Examples
4.3.4 Extension of the Results to Regions in Several Dimensions	Exercises 4.5	A4.5.2 Proof of the Fourier Integral Theorem
Exercises 4.2	Appendix to Chapter IV	A4.6 THE EULERIAN INTEGRALS (GAMMA FUNCTION)
4.4 TRANSFORMATION OF MULTIPLE INTEGRALS	A4.1 THE EXISTENCE OF THE MULTIPLE INTEGRAL	A4.6.1 Definition and Functional Equation
		A4.6.2 Convex Functions: Proof of Bohr's Theorem
4.4.1 Two Dimensions. Remarks	A4.1.1 The Content of Plane Regions and Regions of Higher Dimensions	Exercises 4.8
4.4.2 Regions of more than Two Dimensions	A4.1.2 A Theorem on Smooth Arcs	A4.7 DIFFERENTIATION AND INTEGRATION TO FRACTIONAL ORDER. ABEL'S INTEGRAL EQUATION
Exercises 4.3	A4.1.3 The Existence of the Multiple Integral of a Continuous Function	A4.8. NOTE ON THE DEFINITION OF THE AREA OF A CURVED SURFACE