Product of uniforms

If $U_1, U_2 \overset{iid}{\sim} U[0,1]$, compute $Pr(U_1U_2 \le .5)$?

Solution

\(\begin{aligned} \int_{0}^1\int_{0}^{\min(1,.5/u_1)} du_2 du_1 &= .5 + \int_{.5}^1\int_{0}^{.5/u_1} du_2 du_1 \\ &= \int_{.5}^1 .5/u_1 du_1 \\ &= .5(1 + \ln(2)) \end{aligned}\)