Find the moment of inertia about the $z$-axis of a wire which lies along the circle $x^2 + y^2 = R^2$, with density $δ(x, y) = x^2$, where $R$ is any finite radius.
Here's what I have so far:
$Iz$ = $\int\int_S (x^2+y^2)δd\sigma $
$Iz$ = $\int\int_S (x^2+y^2)x^2|r_u \times r_v|dudv$
I'm not quite sure what to do after this, though. I don't understand how to find $r_u$ or $r_v$, or what I should be setting up as my surface of integration. Would anyone be able to point me in the right direction here?
It is not clear to me why you are doing a surface integral when there is no surface at all. On a curve we can move in two directions, pretty much just like on a wire. On a surface we can move in an infinite amount of directions.
You are interested in computing the line integral,
$$\oint_{x^2+y^2=R^2} (x^2+y^2)\delta ds$$
This is not hard, just parametrize with $x=R \cos (\theta)$ and $y=R \sin (\theta)$ with $\theta \in [0,2\pi]$.