
Expectation
I need to find E[Y/X=x] and E[X/Y=y] for
f(x,y)=24xy (x,y) is region bounded by y=x^2 and y=x.
I'm having trouble deciding on the limits of integration for the two relevant integrals:
yf(y/x)dx and xf(x/y)dy
how do I find the limits for the integrations above?
(I apologise for my lack of LaTex!)

I first checked to see if it's a valid density and it is.
The line y=x is above $\displaystyle y=x^2$ and the points of intersection are (0,0) to (1,1).
$\displaystyle f(x)=24\int_{x^2}^x xydy=12x(x^2x^4)$
and
$\displaystyle f(y)=24\int_y^{\sqrt y} xydx=12y(yy^2)$
Well once you obtain f(xy) and f(yx) 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(xy) and f(yx), then
$\displaystyle E(XY=y)=\int_0^1 x f(xy)dx$
and
$\displaystyle E(YX=x)=\int_0^1 y f(yx)dy$