Double integrals over non-rectangular regions

 Goal: Evaluate a double integral over a nonrectangular region

In the previous post, we stated that, in order to evaluate a double integral over a general, non-rectangular region we need to express this region as a type I or type II, The following theorems allow to reach this goal.

Theorem I

Double integrals over nonrectangular regions

Suppose g(x,y) is the extension to the rectangle R of the integrable function f(x,y) defined on the region D where D is inside R. Then g(x,y) is integrable and we define the double integral f(x,y) over D by:

Theorem II

Example

Solution