ok, in my first post in this thread i tried to simplify things by eliminating z axis but if you want to see things in 3 dimensional here is how it works:

there's no

in the equation

which means we only need to find the cross section of the surface with some

and then rotate the cross section about

axis because

so, for example, let

then you get

so the cross section is the circle

in the plane

which is

xz plane. sketch this circle and then rotate it about

axis. what do you get? a torus of course!

to answer your second question,

is the Jacobian of change of variables from cartisian to spherical coordinates, exactly like

in polar. you always have to put it when calculating the

volume in spherical coordinates. also the tripple integral is over

__the region__ bounded by the surface of the torus and not over the surface of the torus. so

and

__not__