The first thing you have to do is figure out the limits. Whenever you have a region that is complex, its a good idea to see what kind of co-ordinate systems are useful for the problem.
The typical co-ordinate systems used include Cartesian (normal R^n), spherical, cylindrical, and even parabolic:
Parabolic coordinates - Wikipedia, the free encyclopedia
Once you have an appropriate co-ordinate system that allows the easiest way to do integration, the next step is convert your integral to that space and to do that, you need the multivariate integral via substitution which involves using a special term called a Jacobian.
So to get you started, look at parabolic co-ordinate systems, use your constraints to get the limits of integration and then transfer your problem to the appropriate co-ordinate system (through substitution method for multivariate integrals) so that you can evaluate it.