Answer
$$ \bar{x}=\frac{M_{y z}}{M}=\frac{3}{\pi},\ \ \bar{y}=\frac{M_{x z}}{M}=\frac{3}{\pi},\ \ \bar{z}=\frac{M_{x y}}{M}=\frac{3}{4}$$
Work Step by Step
Since
\begin{align*}
M&=\int_{0}^{\pi / 2} \int_{0}^{2} \int_{0}^{r} d z r d r d r d \theta\\
&=\int_{0}^{\pi / 2} \int_{0}^{2} r^{2} d r d \theta\\
&=\frac{8}{3} \int_{0}^{\pi / 2} d \theta\\
&=\frac{4 \pi}{3} \\
M_{y z}&=\int_{0}^{\pi / 2} \int_{0}^{2} \int_{0}^{r} x d z r d r d \theta\\
&=\int_{0}^{\pi / 2} \int_{0}^{2} r^{3} \cos \theta d r d \theta\\
&=4 \int_{0}^{\pi / 2} \cos \theta d \theta\\
&=4\\
M_{x z}&=\int_{0}^{\pi / 2} \int_{0}^{2} y d z r d r d \theta\\
&=\int_{0}^{\pi / 2} \int_{0}^{2} r^{3} \sin \theta d r d \theta\\
&=4 \int_{0}^{\pi / 2} \sin \theta d \theta=4\\
M_{x y}&=\int_{0}^{\pi / 2} \int_{0}^{2} \int_{0}^{r} z d z r d r d \theta\\
&=\frac{1}{2} \int_{0}^{\pi / 2} \int_{0}^{2} r^{3} d r d r d \theta\\
&=2 \int_{0}^{\pi / 2} d \theta=\pi
\end{align*}
$$ \bar{x}=\frac{M_{y z}}{M}=\frac{3}{\pi},\ \ \bar{y}=\frac{M_{x z}}{M}=\frac{3}{\pi},\ \ \bar{z}=\frac{M_{x y}}{M}=\frac{3}{4}$$
