Ok, so solving triple integrals in Cartesian, cylindrical, and spherical is not a problem for me. I rip through solving them no problem. My issue is setting them up. When given a lower bound and an upper bound (normally always given in Cartesian - x,y,z - form) I always seem to screw one of them up, which means I screw up the whole problem. Any tips or tricks to help me see why we use the bounds we do for cylindrical and spherical problems would be greatly appreciated!

Some example problems - all solving for the volume of some region. (I just need the integrals set up and why not solved):

1.) Find the volume of the regionEbounded by the paraboloidsz=x^2+y^2andz= 36 - 8x^2- 8y^2. Use cylindrical coordinates.

2.) Find the volume of the solid that lies within the spherex^2 +y^2 +z^2 = 49, above thexy-plane, and below the following cone.

3.) Evaluate the integral below, whereElies between the spheresx^2 +y^2 +z^2 = 1 andx^2 +y^2 +z^2 = 4 in the first octant.

Any rules of thumb I should follow all the time etc would be greatly appreciated. Usually solving for r or p, depending on if its cylindrical or spherical, is never a problem. But normally at least one of the integrals is 0 - 2pi but I can't seem to find a reasoning as to why...