Chance two uniforms multiply to more than a half

Let $X$ and $Y$ be independent, each uniformly distributed on $(0,1)$. Compute $\mathbb{P}(XY > \tfrac12)$, rounded to two decimal places.

About $0.15$. The region $xy>\tfrac12$ in the unit square needs $x>\tfrac12$ and $y>\tfrac{1}{2x}$, so its area is $\int_{1/2}^{1}\left(1-\tfrac{1}{2x}\right)dx=\tfrac12-\tfrac12\ln 2=\tfrac12(1-\ln 2)\approx 0.153$.