MA 242.003

• Day 36 – February 26, 2013• Section 12.3: Double Integrals over General

Regions

Section 12.3: Double Integrals over General Regions

Problem: Compute the double integral of f(x,y) over the region D shown in the diagram.

Solution:

Section 12.3: Double Integrals over General Regions

Problem: Compute the double integral of f(x,y) over the region D shown in the diagram.

Solution:

Section 12.3: Double Integrals over General Regions

Problem: Compute the double integral of f(x,y) over the region D shown in the diagram.

Section 12.3: Double Integrals over General Regions

Problem: Compute the double integral of f(x,y) over the region D shown in the diagram.

It turns out that if we can integrate over 2 special types of regions, then properties of integrals implies we can integrate over general regions.

Some Examples:

Some Examples:

Some Examples:

Question: How do we evaluate a double integral over a type I region?

Question: How do we evaluate a double integral over a type I region?

Question: How do we evaluate a double integral over a type I region?

Question: How do we evaluate a double integral over a type I region?

Example:

Example:

Example type II regions:

Example type II regions:

Example type II regions:

Example type II regions:

A circular region is type I

Example type II regions:

A circular region is also type II

Using techniques similar to the above we can establish the following:

Using techniques similar to the above we can establish the following:

Treat the region D as type II this time.

(continuation of example)

“Reversing the order of Integration”

“Reversing the order of Integration”

Does NOT mean

“Reversing the order of Integration”

Does NOT mean

“Reversing the order of Integration”

“Reversing the order of Integration”

Step #1: Given an iterated integral over a type I region, for example:

Sketch the region in the xy-plane given by a

“Reversing the order of Integration”

Step #1: Given an iterated integral over a type I region, for example:

Sketch the region in the xy-plane given by a

Step #2: Describe the region as (one or more) type II region(s).

“Reversing the order of Integration”

Step #1: Given an iterated integral over a type I region, for example:

Sketch the region in the xy-plane given by a

Step #2: Describe the region as (one or more) type II region(s).

Step #3: Set up the iterated integral over the type II region(s).

“Reversing the order of Integration”

Step #1: Given an iterated integral over a type II region, for example:

Sketch the region in the xy-plane given by a

Step #2: Describe the region as (one or more) type I region(s).

Step #3: Set up the iterated integral over the type I region(s).

Reversing the order of integration can turn an impossible task into something that is computable.

Reversing the order of integration can turn an impossible task into something that is computable.

Properties of Double Integrals

Properties of Double Integrals

Recall from section 12.1:

Properties of Double Integrals

Properties of Double Integrals