Since even for N= 1, that integral has no elementary formula, I doubt you are going to find anything very useful.
No. This only works where we have the entire first quadrant (and then runs from 0 to not ) so that r will go to infinity for all . Integrating from 0 to b gives a rectangle in the xy-plane and, in polar coordinates, r will run from 0 to either the right or top edge of the rectangle and that r will depend on theta. For r runs from 0 to and for r runs from 0 to .
Furthermore, the upper and lower bounds can be computed explicitly.
Isn't this correct firstly? and if so, can't we extend this to more dimensions?
Thanks.