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).
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