I first checked to see if it's a valid density and it is.

The line y=x is above and the points of intersection are (0,0) to (1,1).

and

Well once you obtain f(x|y) and f(y|x) by division the bounds are easy, it's 0 to 1 in those two integrals.

That leads me to believe your problems are is obtaining the correct conditional densities.

Obtain f(x|y) and f(y|x), then

and