Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii R₁ and R₂.
Figure

Let the mass of the plate be M.
Consider a small semicircular portion of mass dm and radius r, as shown in fig.



$dm=\frac{M\pi rdr}{{\displaystyle \frac{\pi \left(R_2^2-R_1^2\right)}{2}}}=\frac{M}{{\displaystyle \frac{\left(R_2^2-R_1^2\right)}{2}}}rdr$
The centre of mass is given as:
$y_{cm}=\frac{\int ydm}{M}$

$$\begin{gathered} y_{cm}={\int }_{R_1}^{R_2}\left(\frac{2r}{\pi }\right).\frac{M}{{\displaystyle \frac{\left(R_2^2-R_1^2\right)}{2}}}\times \frac{rdr}{M} \\ =\frac{2}{\pi {\displaystyle \frac{\left(R_2^2-R_1^2\right)}{2}}}{\int }_{R_1}^{R_2}{r}^{2}dr \\ =\frac{2}{\pi {\displaystyle \frac{\left(R_2^2-R_1^2\right)}{2}}}\left[\frac{\left(R_2^3-R_1^3\right)}{3}\right] \\ =\frac{4\left(R_1^2+R_1{R}_2+R_2^2\right)}{3\pi \left(R_1+R_2\right)} \end{gathered}$$