Chapter 6.8, Problem 69E

Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must satisfy $xy<\frac{1}{2}$ in order for the area condition to be met. Then argue that the required probability is $\frac{1}{2}+{\displaystyle {\int }_{1/2}^1\frac{dx}{2x}}$ and evaluate the integral.