**Multiple Integrals**

## Double Integrals over Rectangles

26 min 3 Examples

- Double Integrals over Rectangles as it relates to Riemann Sums from Calc 1
- Overview of how to approximate the volume Analytically and Geometrically using Riemann Sums
- Example of approximating volume over a square region using lower left sample points
- Example of approximating volume over a square region using upper right sample points
- Example of approximating volume using the Midpoint Rule

## Average Value and Double Integral Properties

21 min 5 Examples

- Overview of finding the average value in single-variable and multivariable calculus
- Example of finding the average value of a function of two variables
- Overview of the Properties of Double Integrals
- Four examples of evaluating double integrals over rectangles using properties

## Iterated Integrals

34 min 3 Examples

- Overview of how to evaluate a double integral using Fubini’s Theorem
- Demonstration of how to use Fubini’s Theorem
- Example #1 of writing and evaluating an Iterated Integral of a rectangular region
- Example #2 of writing and evaluating an Iterated Integral of a rectangular region using trig
- Example #3 of writing and evaluating an Iterated Integral of a rectangular region using U-Substitution

## Double Integrals over General Regions

1 hr 36 min 9 Examples

- How did we find Area Between Curves in Single-Variable Calculus?
- Example showing the relationship between Single and Double Integrals for finding area
- Example #1 using Iterated Integrals finding Area of a Bounded Region
- Example #2 using Iterated Integrals finding Area over General Region
- Example #3 using Iterated Integrals finding Area over General Region
- Example #4 using Iterated Integrals finding Area over General Region
- Overview and Example of how to Reverse/Switch the Order of Integration
- Example #2 of how to Reverse/Switch the Order of Integration
- Example #3 of how to Reverse/Switch the Order of Integration
- Application Example of Double Integrals over General Regions to find Volume

## Double Integrals in Polar Coordinates

1 hr 5 min 5 Examples

- Overview and formula for how to calculate double integrals in polar coordinates
- Example #1 of changing to polar coordinates and evaluating the double integral
- Example #2 of changing to polar coordinates and evaluating the double integral using U-Substitution
- Example #3 of changing to polar coordinates and evaluating the double integral using U-Substitution
- Example #4 of evaluating a double integral in polar coordinates using a half-angle identity
- Example #5 of finding the volume of a solid in polar coordinates

## Applications of Double Integrals: Density, Mass and Moments of Inertia

1 hr 1 min 4 Examples

- Overview of Mass, Density, Moments, and the Center of Mass for Double Integrals
- Example of how to find the Mass and Center of Mass using Double Integrals
- Example of finding Total Charge given charge density at a point
- Overview of Moments of Inertia (Second Moments)
- Example of finding the Moments of Inertia
- Overview of Probability Density
- Example of finding the Probability Density for a Joint Density Function

## Surface Area

38 min 3 Examples

- Overview of Surface Area using Double Integrals
- Example #1 of finding Surface Area of the part of a plane
- Example #2 of finding Surface Area of a cylinder
- Example #3 of finding Surface Area using Polar Coordinates

## Triple Integrals

44 min 3 Examples

- Overview of Triple Integrals
- Example #1 Evaluating a Triple Integral over a rectangular box
- Example #2 Evaluating a Triple Integral over a general bounded region
- Example #3 Evaluating a Triple Integral over a general bounded region

## Triple Integrals in Cylindrical Coordinates

43 min 3 Examples

- Overview and Formula for Triple Integrals in Cylindrical Coordinates
- Example #1 Evaluating a Triple Integral in Cylindrical Coordinates
- Example #2 Evaluating a Triple Integral in Cylindrical Coordinates
- Example #3 Converting a Triple Integral from Rectangular to Cylindrical

## Triple Integrals in Spherical Coordinates

37 min 3 Examples

- Review of relationship between Rectangular and Spherical Coordinates
- Formula for Triple Integrals in Spherical Coordinates
- Example #1 Evaluating a Triple Integral in Spherical Coordinates
- Example #2 Evaluating a Triple Integral in Spherical Coordinates
- Example #3 Converting a Triple Integral from Rectangular to Spherical

## Change of Variables in Multiple Integrals

1 hr 6 min 7 Examples

- Overview of Change of Variables and Transformations
- Example #1 Determining a new region by applying a given transformation
- Example #2 Determining a new region by applying a given transformation
- Overview of the Jacobian Transformation
- Example #1 Finding the Jacobian Determinate for two variables
- Example #2 Finding the Jacobian Determinate in polar coordinates
- Example #3 Finding the Jacobian Determinate for three variables
- Overview and Formula for finding the Change of Variables for Multiple Integrals
- Example #1 Evaluating a double integral given an appropriate change of variables
- Example #2 Evaluating a double integral given an appropriate change of variables