How do I know which limits to place on a triple integration problem?

Question

I'm doing homework/studying for a test right now and I think the reason I am not getting right answers is the limits I choose for triple (and some double) integration problems.

For example...one problem is: evaluate the triple integral - f(x,y,z)= x+y; T is the region between the surfaces z=2-x^2 and z=x^2 for 0<=y<=3 . The limits I put on it were: y: 3 to 0; x: sqrt(2) to 0 (my reasoning for this is from using the z=2-x^2 equation and evaluating it for z=0...is this valid?); and z: 2-x^2 to x^2 ...If these are right, then I'm making computational errors, but I get it. If these are not right, then I do not get it.

Any light you could shed on either the problem or determining the limits would be very much appreciated. By the way, the right answer to that problem is 12. Thanks.

Answer 1

Graph z=2-x² and z=x² in a plane. These are two intersecting parabolas. Find the intersection points by setting 2-x² = x². The solutions give you the limits on x.

This is a somewhat atypical situation. Here the surface equations don't depend on y, so the limits on x do not depend on y.