By X+X you mean 2X, well that is U(0,2).

For W=X+Y, wher you need independence, the classic way to do is solve via the CDF of W.

Draw the unit square (0,1)X(0,1) and find F(w) where 0<w<1 and 1<w<2.

The derivative will give you the density which is

f(w)=w on (0,1) and f=2-w on [1,2) and 0 elsewhere.

IS through=throw?