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So the only time that you want to evaluate an integral representing the volume of $D$ is when it is specifically asked. For example question #$4e$ off of the in-class worksheet from July 24th. Otherwise, always use the function given and convert it to whatever coordinate system you are going to use. So for this problem it would be: $$\int_0^{2\pi}\int_0^{\pi}\int_0^1 (\rho^2)\rho^2 \sin(\phi) d\rho d\phi d\Theta$$