For the figure there is an arc of a thin ring, that subtends at an angle $\theta$ at the centre. The mass of the arc is $m$ and its radius it $R$. Locate the centre of mass of the arc

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 From symmetry, it can be said that the COM lies on Y-axis. Mass of the element shown is $dm=\frac{m}{\theta }d\alpha .$

y-coordinate of element is

$$\begin{gathered} y=R \sin \left(\alpha +90°-\frac{\theta }{2}\right) \\ \therefore Y_{com}=\frac{\int ydm}{\int dm}=\frac{{\int }_0^{\theta }R\sin \left(90°+\alpha -\frac{\theta }{2}\right)\frac{m}{\theta }d\alpha }{m} \\ =\frac{R}{\theta }{\int }_0^{\theta }\cos \left(\alpha -\frac{\theta }{2}\right)d\alpha \\ =\frac{R}{\theta }{\left[\sin \left(\alpha -\frac{\theta }{2}\right)\right]}_0^{\theta } \\ =\frac{R}{\theta }·2\sin \theta /2 \end{gathered}$$