Solving Problems in Mathematical Analysis, Part II: Definite, Improper and Multidimensional Integrals, Functions of Several Variables and Differential Equations (Problem Books in Mathematics)

Preface
Contents
Definitions and Notation
1 Exploring the Riemann and Definite Integral
	1.1 Examining the Integrability of Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	1.2 Finding Riemann Integrals by Definition
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	1.3 Finding Limits of Certain Sequences
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	1.4 Calculating Various Interesting Definite Integrals
		Problem 1
		Solution
		Problem 2
		Solution
			Integral (a)
			Integral (b)
			Integral (c)
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
		Problem 6
		Solution
	1.5 Explaining Several Apparent Paradoxes
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	1.6 Using the (Second) Mean Value Theorem
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	1.7 Exercises for Independent Work
2 Examining Improper Integrals
	2.1 Investigating the Convergence of Integrals by Definition
		Problem 1
		Solution
		Problem 2
		Solution
	2.2 Using Different Criteria
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	2.3 Using Integral Test for Convergence of Series
		Problem 1
		Solution
		Problem 2
		Solution
	2.4 Exercises for Independent Work
3 Applying One-Dimensional Integrals to Geometry and Physics
	3.1 Finding Lengths of Curves
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	3.2 Calculating Areas of Surfaces
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	3.3 Finding Volumes and Surface Areas of Solids of Revolution
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	3.4 Finding Various Physical Quantities
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	3.5 Exercises for Independent Work
4 Dealing with Functions of Several Variables
	4.1 Finding Images and Preimages of Sets
		Problem 1
		Solution
		Problem 2
		Solution
	4.2 Examining Limits and Continuity of Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	4.3 Exercises for Independent Work
5 Investigating Derivatives of Multivariable Functions
	5.1 Calculating Partial and Directional Derivatives
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	5.2 Examining Differentiability of Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	5.3 Exercises for Independent Work
6 Examining Higher Derivatives, Differential Expressions, and Taylor's Formula
	6.1 Verifying the Existence of the Second Derivatives
		Problem 1
		Solution
		Problem 2
		Solution
	6.2 Transforming Differential Expressions and Operators
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	6.3 Expanding Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	6.4 Exercises for Independent Work
7 Examining Extremes and Other Important Points
	7.1 Looking for Global Maxima and Minima of Functions on Compact Sets
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	7.2 Examining Local Extremes and Saddle Points of Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	7.3 Exercises for Independent Work
8 Examining Implicit and Inverse Functions
	8.1 Investigating the Existence and Extremes of Implicit Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	8.2 Finding Derivatives of Inverse Functions
		Problem 1
		Solution
		Problem 2
		Solution
	8.3 Exercises for Independent Work
9 Solving Differential Equations of the First Order
	9.1 Finding Solutions of Separable Equations
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	9.2 Solving Homogeneous Equations
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	9.3 Solving Several Specific Equations
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	9.4 Solving Exact Equations
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	9.5 Exercises for Independent Work
10 Solving Differential Equations of Higher Orders
	10.1 Solving Linear Equations with Constant Coefficients
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	10.2 Using Various Specific Methods
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	10.3 Exercises for Independent Work
11 Solving Systems of First-Order Differential Equations
	11.1 Using the Method of Elimination of Variables
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	11.2 Solving Systems of Linear Equations with ConstantCoefficients
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
		Problem 5
		Solution
	11.3 Exercises for Independent Work
12 Integrating in Many Dimensions
	12.1 Examining the Integrability of Functions
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	12.2 Calculating Integrals on Given Domains
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	12.3 Changing the Order of Integration
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	12.4 Exercises for Independent Work
13 Applying Multidimensional Integrals to Geometry and Physics
	13.1 Finding Areas of Flat Surfaces
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	13.2 Calculating Volumes of Solids
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
		Problem 4
		Solution
	13.3 Finding Center-of-Mass Locations
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	13.4 Calculating Moments of Inertia
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	13.5 Finding Various Physical Quantities
		Problem 1
		Solution
		Problem 2
		Solution
		Problem 3
		Solution
	13.6 Exercises for Independent Work
Index