Bounds for the PDF of $(X,XY)$ if the PDF of $(X,Y)$ has support $0\leq X\leq Y\leq1$ 
Given the PDF $\,f_{XY}(x,y)=8xy,\, 0\leq x \leq y \leq 1$. Let $U=X$ and $V=XY$. Find the pdf of $(U,V)$.

My answer:
So, $X=U$ and $Y= V/U$, The Jacobian J of the inverse transformation would then equal:
$J=\frac{1}{u}$..., then 
How do I find the bounds for the integration? of $f_{U,V}(u,v)\int_?\int_? F_{X,Y}(u,v/u)|J|dudv$

$U=X$ so the range for $U$ is $0\lt U\lt 1$.
$V=XY$ has its maximum when $Y=1$, giving $V=X=U$. $V$ has its minimum when $Y=X$, giving $V=X^2=U^2$.
So the joint pdf for $U,V$ is
$$f_{U,V}(u,v) = |J|f_{X,Y}(x(u,v),\;y(u,v)) = \dfrac{8v}{u},\quad 0\lt u\lt 1\; \text{ and } u^2\lt v\lt u.$$
