I have been searching everywhere online on how to approach these questions. But every example I find, it is said in the question what the density function is. That is what I'm struggling to find with this question, which is as follows:
The density $\rho$ (mass per unit area) of a semi-circular lamina $\Omega$ of radius a is proportional to the distance from the centre of the circle. Find the centre of mass $(\bar{x},\bar{y})$ of the lamina, taking $\bar{x} = 0$ by symmetry, given $$ m\bar{y} = M_y = \iint_\Omega y\rho(x,y)\mspace{4mu}dA \quad \text{and} \quad m = \iint_\Omega \rho(x,y)\mspace{4mu}dA $$ where $m$ is the mass of the lamina. Use Planar polar coordinates.
I am trying very hard to get better at these questions so a detailed explanation would be very much appreciated.
Thank you