Answer
$\bar x=\bar y=0$
Work Step by Step
We calculate the center of mass as follows:
First calculate the mass:
$M=4\int^{\pi/2}_0 \int^1_0 \int^r_0 $ dz r dr $ d\theta $
=$4\int^{\pi/2}_0 \int^1_0 r^2$ dr $ d\theta $
=$\frac{4}{3}\int^{\pi/2}_0 d\theta $
=$\frac{2\pi}{3}$
Now calculate the moment:
$M_{xy}$= $\int^{2\pi}_0 \int^1_0 \int^r_0 $ z dz r dr $ d\theta $
=$\frac{1}{2}\int^{2\pi}_0 \int^1_0 r^3 dr d\theta $
=$\frac{1}{8} \int^{2\pi}_0 d\theta $
=$\frac{\pi}{4}$
And take the ratio:
$\frac{M_{xy}}{M}=(\frac{\pi}{4})(\frac{3}{2\pi})=\frac{3}{8}$
Thus, we have (by symmetry):
$\bar x=\bar y=0$
