I think an easier approach is:

\(\displaystyle E(|X-Y|) =\int \int |X-Y|f_{X,Y} dxdy\)

Assuming......

\(\displaystyle X\) ~ \(\displaystyle \text{Uniform(0,1);}\) \(\displaystyle Y\) ~ \(\displaystyle \text{Uniform(0,1) i.i.d.}\)

......makes this fairly simple, actually.