Here's the deal: When you transform a double intergral that goes over a set D < RxR bounded on y-axes by g1(x) and g2(x) in two "normal" ones(litteral translation from my language would be subsequent integrals - don't know the word in English) how do you swap the integrals by x and by y(taking y from y1=const to y2=const and x from h1(y) to h2(y)) without visualising the surface itself? In other words is there any recipe for this kind of transformation or is it always done ad hoc i. e. drawing a picture which you try to figure out?
The English phrase is "repeated integral" or "iterated integral". There is no general formula for changing the limits of integration. Depending on the specific regions, the integral can be MUCH more complicated one way than the other.