Moment of Interita about x-axis in Polar Coordinates

[jegues]

Homework Statement

A plate with constant mass per unit area $$\rho$$ is bounded by the curve $$(x^{2} +y^{2})^{2} = 9(x^{2} - y^{2})$$ . Find its moment of inertia about the x-axis.

Homework Equations

The Attempt at a Solution

Okay well first I plugged in,

$$x=rcos\theta,y=rsin\theta$$

into my equation and simplified a little giving me the following result,

$$r^{2} = 9cos(2\theta)$$

Now I know the moment of inertia about the x-axis is defined to be,

$$\int \int_{R} y^{2} \rho dA$$

Now if I want to consider this in polar coordinates would it simply be,

$$\int^{\beta}_{\alpha} \int^{r_{2}}_{r_{1}} r^{2}sin^{2}\theta \rho rdrd\theta$$

I'm a little confused on how the bounding curve plays into this problem.

Any help?

 

[jegues]

Alright I've done some work on the question and basically I've got up to this point in my integration and I get stuck.

This problem is really only an "evaluate the integral" problem at this point.

Does anyone have any suggestions how to compute the rest of this integral? (If it's correct up to that point)