are independent RV ~U(0,1) (which appears to be what the question is saying,
then the distribution of Y=X1+X2 is the symetric triangular distribution on (0,2).
I can see no way of reconciling this with
p(y) = 1 - 1/2(2 - y)^2
wherever I put brackets in this expression.
The attachment shows the histogram of 100000 realisations of y, and this
is clearly the triangular distribution.