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More specifically, if I have z = f(x,y) and I want to integrate over some domain in the xy plane then I want to say the integral would look like [itex]\int[/itex][itex]\int[/itex] f(x,y) dx dy. Similarly, if it was in polar coords, f(r,theta) then I would have [itex]\int[/itex][itex]\int[/itex] f(r,theta)r dr d theta. I think this is true since the integral is "adding" little boxes of volume z dx dz or z r dr dtheta respectively.

However, if it is a triple integral, whether in spherical, cylindrical or Cartesian, the function is not included in the integral because dV is the little units of volume. Is this correct?