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I agree with Gear Junky. I believe that zero is the correct answer. I would like to offer a geometric perspective on this problem that does not require calculus (but can be easily verified using calculus by following the steps that Gear Junky took). Sometimes you can save a lot of work—and panic—by thinking visually and this is a good example of such a problem.

I have provided an image of the graph of the equations in question for reference. We want to find the area within the blue cylinder that is not taken up by the cones.

The side view of this graph looks like this (this is the $xz$ plane):

You will notice that $z$ ranges from $-r$ to $r$, as Gear Junky said. Both this graph and the integrand ($z$) has symmetry about the $xy$ plane, and for every value above the $xy$ plane (positive $z$), there is an equal but opposite value below the plane. This means that the integral cancels itself and gives us our scary yet beautiful answer of $0$. Here is a view that illustrates the symmetry: