Answer
$\overline {x}=0; \overline{y}=\dfrac{a(k+2)}{2k+\pi}$
Work Step by Step
We notice that the center of mass is symmetric about the y- axis, so $\overline {x}=0$
and $\overline{y}=\dfrac{ m_x}{m}=\dfrac{\int_0^{\pi} a \sin \theta \delta a d\theta}{\int_0^{\pi} \delta a d\theta} $
or, $=\dfrac{\int_0^{\pi} a^2 [1+k|\cos|\theta |] \sin^2 \theta d\theta}{\int_0^{\pi} a [1+k|\cos|\theta |] \sin \theta d\theta} $
or, $=\dfrac{a (-\cos \theta- k \cos (2 \theta)]_0^{\pi/2}}{[\theta +k \sin \theta]_0^{\pi/2}} $
or, $\overline{y}=\dfrac{a(k+2)}{2k+\pi}$
