Conditional Joint Distributions

I am given random variables x and y where $f(x,y) = 1$ for $0 <= x, y <= 1$ and 0 elsewhere. I am asked to find $E(X | Y > X)$.

Here’s what I have. I know I made a mistake somewhere or am misunderstanding something so any help would be appreciated.

I get $f_{x|y}(x, y > frac{1}{2}) = frac{f_{xy}(x, y > x)}{f_{y}(y > x)}$.

I get $f_{xy}(x, y > x) = int_{0}^{1} int_{x}^{1} 1 {,dy} {,dx} = int_{0}^{1} (1-x) {,dx} = 1/2$

I also get ${f_{y}(y > x)} = int_{x}^{1} 1 {,dy} = 1-x$

However, that means $f_{x|y}(x, y > frac{1}{2}) = frac{1}{2(1-x)}$.

This can’t be right since when I plug this into $E(X | Y > X) = int_{0}^{1} x * frac{1}{2(1-x)} {,dx}$, this is an undefined value…so clearly I messed up. What step am I doing wrong? Or am I not even approaching the problem in the right way?