Probability. Define Upper bound and Lower bound for double integral.

Question

I am finding trouble of finding bounds for double integral for statistic. Here the question looks like: given joint pdf = $\frac{3x}4$, $0<x<1,\;0<y<4x$.

I have to find pdf of $R=XY$. the solution is $1-P(Y>\frac rX)$ where $$P(Y>\frac rX) =\int_{\frac{\sqrt r}2}^1\int_{\frac rx}^{4x}f(x,y)\cdot dxdy$$

however, I do not get why the lower bound of first integral would be $\frac{\sqrt r}2$.

please help me.

Answer 1

You intended to write $dydx$ rather than $dxdy$.

Draw out the region of interest, $\frac{\sqrt{r}}2$ is the $x$-coordinate of the intersection $xy=r$ and $y=4x$.