Answer
$\bar x=\bar y=0$
Work Step by Step
We need to find center of mass as follows:
$$ M=4 \times \int^{\pi/2}_0 \int^1_0 \int^r_0 \space dz \space r dr d\theta =\dfrac{4}{3}\int^{\pi/2}_0 d\theta =\dfrac{2\pi}{3}$$
$ M_{xy}= \int^{2\pi}_0 \int^1_0 \int^r_0 z dz r dr \space d\theta =\dfrac{1}{2}\int^{2\pi}_0 \int^1_0 r^3 \space dr \space d\theta =\dfrac{\pi}{4}$
Now, $\dfrac{M_{xy}}{M}=(\dfrac{\pi}{4})(\dfrac{3}{2\pi})=\dfrac{3}{8}$
Using symmetry, we have:
$\bar x=\bar y=0$
