Improper Double Integrals

 Definition of an improper integral

It's preferable to deal with improper integrals of functions over rectangles or simple regions where these functions have finitely many discontinuities. However, not such improper integrals can be evaluated. A form of Fubini's theorem allows to evaluate some types of improper double integrals.

Fubini's theorem for improper integrals

Two conditions are necessary for the theorem to work. The function has to be nonnegative on D and has many finitely discontinuities inside D.

Example

Solution

Let's start by plotting the region. 

The function f is continuous on all points of the region D except in (0,0). If you keep the expression of the region, it would be difficult to calculate the double integral. However, if we express the region in 

the following way, the double integral becomes easy to calculate.

Practice

Consider the function f(x,y) = sin(y) /y over the region: