Answer
$$ \bar{z}=\frac{M_{x y}}{M}=(\pi)\left(\frac{3}{8 \pi}\right)=\frac{3}{8},\ \bar{x}=\bar{y}=0 $$
Work Step by Step
Since
\begin{align*}
M&=\frac{8 \pi}{3} \\
M_{x y}&=\int_{0}^{2 \pi} \int_{\pi / 3}^{\pi / 2} \int_{0}^{2} z \rho^{2} \sin \phi d \rho d \phi d \theta\\
&=\int_{0}^{2 \pi} \int_{\pi / 3}^{\pi / 2} \int_{0}^{2} \rho^{3} \cos \phi \sin \phi d \rho d \phi d \theta\\
&=4 \int_{0}^{2 \pi} \int_{\pi / 3}^{\pi / 2} \cos \phi \sin \phi d \phi d \theta\\
&=4 \int_{0}^{2 \pi}\left[\frac{\sin ^{2} \phi}{2}\right]_{\pi / 3}^{\pi / 2} d \theta\\
&=4 \int_{0}^{2 \pi}\left(\frac{1}{2}-\frac{3}{8}\right) d \theta\\
&=\frac{1}{2} \int_{0}^{2 \pi} d \theta=\pi
\end{align*}
Then by symmetry
$$ \bar{z}=\frac{M_{x y}}{M}=(\pi)\left(\frac{3}{8 \pi}\right)=\frac{3}{8},\ \bar{x}=\bar{y}=0 $$
