Q:

Finding probability that the function of a random variable is less than another random variable

X and Y are random variables, whose joint density function is $f(x,y)$ for $\infty<x<\infty$ and $\infty<y<\infty$. I am trying to find $P[X^2<Y]$.
Here's how I plan on solving the problem:
1) Find the marginal density function $f_X(x)$. 
2) Let $Z=X^2$ and find the marginal density function $f_z(z)$. 
3) Find the joint density function of Z and Y, over which I can integrate over the area $Z-Y=0$. 
However, I am stuck on step 3. Since I am not told whether X and Y are independent to begin with, I do not know to find the joint density function of Z and Y. 
Thoughts? 

A:

The proposed strategy is far too complicated. Let $D$ be the region above the curve $y=x^2$. You want
$$\iint_D f(x,y)\,dy\,dx.$$
This in a sense finishes things if we do not know $f$. You can express the integral, if you wish, as an iterated integral, $y$ goes from $x^2$ to $\infty$, $x$ then goes from $-\infty$ to $\infty$. 

