Double Integrals · Superchallenge problem

Numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0, 1)$. What is the probability that the integer nearest to $\dfrac{x}{y}$ is even? (Count $0$ as even.) You may use $1 - \dfrac13 + \dfrac15 - \dfrac17 + \cdots = \dfrac{\pi}{4}$. Give an exact answer.