General regions of Integration

 Objective: recognize when a function f(x,y) is integrable over a general region.

General regions of integration 

Let's consider the following figure:

Here the bounded region D is enclosed by a rectangular region in the plane. Let's suppose a function z = f(x,y) defined over the general planar region D. In order to define integrals of f over the region D, we need to extend the definition of f to include all the points of the rectangular region R. In order to do so, we introduce a new function g defined as follow:

We assume that the region D is a piecewise smooth and continuous function. The function g should be integrable over the region R. This happens as long as the region D is bounded by simple curves. We consider two types of planar bounded regions as stated in the following definition:

Definition

Example

Solution

Practice