Finding moment of Inertia of area bounded by a polar equation

Question

I have to find Moment of Inertia of area bounded by curve $r^2 = a^2 \cos{2 \Theta}$

Since I already know the shape of given curve, I can easily take an element area and double integrate using suitable values of $r$ and $\Theta$.

But is there any general method to follow, say in an examination, when I don't know what the curve of given polar equation looks like? Finding $r$ values corresponding to specific $\Theta$ seems time consuming.

Or, does there exist any method for finding Moment of Inertia which does not require plotting of curve?

Answer 1

HINT: the curve $r^2=a^2\cos(2\theta)$ is the lemniscate of Bernoulli in polar coordinates.