Let $X$ and $Y$ be random variables with joint pdf
$$f_{XY}(x,y) = \begin{cases} 1, &0 < x <1,\ \ x < y < x+1 \\ 0, & \text{otherwise} \end{cases}$$
Find $P(XY \le a)$
I am going to solve this using the CDF method with integration.
Based on the information given, my setup is the following:
$$0 < x < y < 1 \\ x < y < x+1 < 2 \\ xy < a$$
I will make a diagram, where the blue curve represents an example of $0<a<1$, and the green curve represents an example of $1<a<2$:
For $0<a<1$, I will integrate:
$$\int_0^a \int_0^{\min \{{a \over y}, \ y\}}\, dx \ dy $$
For $1<a<2$, I will integrate:
$$\int_0^2 \int_0^{\min \{{a \over y},\ 1 \}}\, dx \ dy $$
My question is twofold:
Are my steps correct so far, and if not, where am I going wrong?
Is there a simpler way to do this?