I'm trying to compute the moment of inertia of a 2D ellipse about the z axis, centered on the origin, with major/minor axes aligned to the x and y axes. My best guess was to try to compute it as:
$4\rho \int_0^a \int_0^{\sqrt{b^2(1 - x^2/a^2)}}(x^2 +y^2)\,dydx$
... I couldn't figure out how to integrate that. Is there a better way or a trick, or is the formula known? I'd also be happy with a good numerical approximation given a and b.