If I have two independent uniforms $U_1$ and $U_2$, one with parameter $(a_1, a_2)$, the other one with parameters $(b_1, b_2)$ and I want to find out the variance of $U = U_1 + U_2$, I use
$Var(U) = Var(U_1) + Var(U_2) +Cov(U_1,U_2) = \frac{(a_2-a_1)^2}{12} +\frac{(b_2-b_1)^2}{12} +0$
Is it sound to assume the covariance is 0, since the the R.V.'s are independent, and therefore uncorrelated, or may I not assume this?