Example 1 :
Change the order of integration.
The given integral is
Comparing the two integrals, the curves bounding the region of integration are given by
Let us trace all the curves to get region of integration
After changing the order of integration, the integral changes to
The domain has to be partitioned by line y = a into three regions as shown in figure.
For region I, x varies from to and y varies from y = 0 to y = a
For region II, x varies from to x = 2a and y varies from y = 0 to y = a
For region III, x varies from to x = 2a and y varies from y = a to y = 2a
So, after changing the order, the integral becomes