When to use spherical and cylindrical coordinates?

[nb89]

For example with a paraboloid, which do i use? I am also slightly confused with the limits in the integral. If doing a triple integral with drdθdΦ i understand the limits of the dr integral but when it comes to dθ and dΦ i don't understand why sometimes its 0 to 2π or 0 to π etc.
For example with this case: a paraboloid that has the following equation z = x^2 + y^2
and z <= 9
What i tried doing is using cylindrical (dont really know why), with dz having limits 0 to 9, but i don't get what dr and dθ would be.

 

[slider142]

Draw a picture. Cut the paraboloid into planes where z is a constant. Then you find that r is a function of t (or is a constant), like a polar coordinate system, and z because we are dealing with a series of planes. They should be a series of circles or different radii on each plane. A cylindrical coordinate system is useful when you can slice objects up into planes on an axis so that each plane has a simple polar plot (ie., circles or ellipses).
Always start with a diagram.