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**Laurent** Because $\displaystyle Y_1$ can be close to 2 and $\displaystyle Y_2$ close to 0 simultaneously, we must expect that the density of $\displaystyle U$ is positive near 2 (from below).

If $\displaystyle 1\leq u\leq 2$, the lines intersect differently: the line $\displaystyle y_1=y_2+u$ meets $\displaystyle y_1=2$ at $\displaystyle y_2=2-u$, so that (look at the graph): $\displaystyle P(Y_1\leq Y_2+u)=\int_0^{2-u}\int_{2y_2}^{y_1+u}dy_1\,dy_2+\int_{2-u}^1\int_{2y_2}^2 dy_1\,dy_2$.

And $\displaystyle P(U\leq u)=1$ if $\displaystyle u>2$, $\displaystyle P(U\leq u)=0$ if $\displaystyle u\leq 0$.