Answer
$\overline {x}=0; \overline {y}=\dfrac{a \pi}{4}$
Work Step by Step
We have a center of mass that is symmetric about the y- axis, which implies that $\overline {x}=0$
We are given that $\delta =k \space \sin \theta$
$\overline{y}=\dfrac{ m_x}{m}=\dfrac{\int_0^{\pi} a \sin \theta \delta a d\theta}{\int_0^{\pi} \delta a d\theta} \\=\dfrac{\int_0^{\pi} a^2 k (\sin^2 \theta d\theta)}{\int_0^{\pi} a \space k (\sin \theta d\theta)} \\=\dfrac{a (\theta-\sin (\dfrac{2 \theta}{2})]_0^{\pi}}{(2)(2)} \\=\dfrac{a \pi}{4}$
