M408M Learning Module Pages
Main page
Chapter 10: Parametric Equations and Polar Coordinates
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives

### Chapter 15: Multiple Integrals

#### Learning module LM 15.1: Multiple integrals:

Background
What is a double integral?
Volumes are double integrals

#### Learning module LM 15.2: Multiple integrals over rectangles:

One variable at a time!
Fubini's Theorem
Notation and order

#### Learning module LM 15.3: Double integrals over general regions:

Type I and Type II regions
Examples
Order of integration
Area and volume revisited

#### Learning module LM 15.4: Double integrals in polar coordinates:

$dA=r\; dr\; d\theta$
Examples of polar integrals

#### Learning module LM 15.5a: Multiple integrals in physics:

Mass, center of mass, and moment of inertia
Triple integrals in physics

#### Learning module LM 15.5b: Integrals in probability and statistics:

Single integrals in probability
Double integrals in probability

#### Learning module LM 15.10: Change of variables:

Change of variable in 1 dimension
Mappings in 2 dimensions
Jacobians
Examples
Cylindrical and spherical coordinates