Understanding Domains

Question

Working on a problem involving a distribution for $f(x,y) = \frac{c}{(x^2+y^2)^2}$ for $x^2+y^2\geq 1$. I thought to make this a well-defined probability distribution to convert to polar, and got $c=\frac{3}{2}$. However, I have been trying to visualize this function with regard to the marginal distributions of $X$ and $Y,$ and have become really confused. When I start to integrate, say for the marginal of $X,$ I expect my domain to be $x^2 \geq \sqrt{y^2-1}$. However, this gives a bizarre result as then the domain of Y is limited to $[-1,1]$. Is this correct? Any help on getting to the marginal distributions would be appreciated.