MA Day 36 – February 26, 2013 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:

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

Example:

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:

Treat the region D as type II this time.

(continuation of example)

Reversing the order of Integration

Does NOT mean

Reversing the order of Integration Does NOT mean

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.

Properties of Double Integrals

Recall from section 12.1:

Properties of Double Integrals