Moment of Inertia of a Lamina Around the Center of Mass

Question

I have a two-dimensional lamina in the $xy$-plane, and I need to calculate the moment of inertia around the center of mass. I know that the moment of inertia around the $x$-axis is $I_x = \int\int y^2 \delta(x,y) \mathrm{d}A$ and $I_y = \int\int x^2 \delta(x,y) \mathrm{d}A$ for the $y$-axis, but I'm not sure how to apply that to / find the formula for the vertical axis going through the center of mass...

Thank you in advance

Answer 1

Not a full answer (so please don't downvote), but this figure should help. The black arrow is the $z$ axis through the center of mass, and the blue arrow represents $r$. Integrate over all possible $r$s of

$$I_z = \iint\limits_{\Omega} r^2\ dm = \iint\limits_{\Omega} (x^2 + y^2)\ dm$$

where $\Omega$ is the laminar region (area).