Correct0<x<1

E(Y|x)=\int^{x} _{-x} \frac{y}{2x}dy=\frac {y^2}{4x} |^{x} _{-x}=\frac{x^2-x^2}{4x}=0???

if you know that Y = y, then we have the restriction -x<y<x as well as 0<x<1.What are the limits of integration for the marginal pdf of Y??

f_2(y)=\int ^?_? f(x,y)dx

, and together with x>y this gives x>|y|.

So the limits are |y|<x<1.