Math Help - moments of inertia

1. moments of inertia

Hi, I needed some help with an integration problem in polar coordinates:

The region R is bound by the polar curve r = 1 - cos(theta).

Set up (do not solve) an iterated integral which gives the moments of inertia about the y axis for the region R. The density is d= x^2 + y^2

Thanks!!!

2. Originally Posted by intrepidy
Hi, I needed some help with an integration problem in polar coordinates:

The region R is bound by the polar curve r = 1 - cos(theta).

Set up (do not solve) an iterated integral which gives the moments of inertia about the y axis for the region R. The density is d= x^2 + y^2

Thanks!!!
In Cartesian coordinates:

$M_y = \int \int_{R} x \, \rho(x, y) \, dx \, dy$.

Switch to polar coordinates:

$M_y = \int \int_{R} r \cos \theta \, \rho(r, \theta) \, r\, dr \, d\theta = \int_{0}^{2 \pi} \int_{0}^{1 - \cos \theta} r^3 \, dr \, d\theta$.