Let $X, Y$ be jointly uniformly distributed random variables, with joint PDF: $$f(x,y) = \frac{1}{25\pi}\mathbf 1_{x^2 + y^2 \leq 25}$$ The question is to compute $P(|XY| \leq \sqrt{5})$, and $P(|X+Y| \leq 5)$.
I believe I have the second one correct, but for the first one I get an extremely complicated answer.
What I did was I drew the disk with radius $5$, and then drew the graph of $|XY| \leq \sqrt{5}$, and found the area that was overlapping between the two, and multiplied it by the joint PDF. But like I said, I got an extremely complicated answer, and was wondering if I was doing this correctly. If someone could help me out it would be appreciated.
The way I attempted to solve this was the following:
I found the area between the disk and the graph $|XY| \leq \sqrt5$ in the first quadrant by integration. Then I subtracted this from the area of the quarter circle in the first quadrant. This gave me the area that I required in the first quadrant. Then I multiplied that area by 4 to get the whole region, then I multiplied it by the joint PDF. I know there's no numbers here, but it came out to be really confusing and long, I just wanted someone to check on this and see if I was doing it correctly.