Joint and marginal distributions of independent uniformly distributed variables 
Suppose that $X_1$ and $X_2$ are independently uniformly distributed on the interval (0,1). Find the joint and marginal distributions of $U=X_1X_2$ and $V=X_1/X_2$.

I think that $f_U(u) = \int^1_0f_{X_1}(x_1)f_{X_2}(u/x_1){1\over{|x_1|}} dx_1$
I'm not sure if that is right. If it is right, would the marginal distribution just be $\int^1_0 X_1X_2 dx_1$ and $\int^1_0 X_1X_2 dx_2$?
Thanks for your help.
 A: The joint PDF of $U$ and $V$ is
$$
f_{U,V}(u,v)=f_{X_1,X_2}(x_1,x_2)\cdot|J|=\frac1{2v}.
$$
Now for the regions. You have $0\le x_1\le1$, this region is corresponding to 
$$0\le \sqrt{uv}\le1\;\Rightarrow\;0\le uv\le 1\;\Rightarrow\;0\le v\le \frac1u.$$ 
It's due to $X_1=\sqrt{UV}$. You also have $0\le x_2\le1$, this region is corresponding to 
$$0\le \sqrt{\dfrac uv}\le1\;\Rightarrow\;0\le \dfrac uv\le1\;\Rightarrow\;0\le u\le v.$$
It's due to your transformation $X_2=\sqrt{\dfrac UV}$. If you plot those regions (check the plot here), you can obtain the marginal PDFs of $U$ and $V$ as follow
$$
\begin{align}
f_U(u)&=\int_{v=u}^{\frac1u}f_{U,V}(u,v)\ dv\ ;\quad\text{the region is bounded by $v=u$ and $v=\frac1u$ if you see it from $u$-axis}\\
&=\int_{v=u}^{\frac1u}\frac1{2v}\ dv\\
&=\left.\frac 12\ln v\right|_{v=u}^{\frac1u}\\
&=\frac 12\ln \left(\frac1u\right)-\frac 12\ln u\\
&=-\ln u\;;\quad\text{ for } 0\le u\le1.
\end{align}
$$
and
$$
\begin{align}
f_V(v)&=\int_{u=0}^{v}f_{U,V}(u,v)\ du\ ;\quad\text{the region is bounded by $u=0$ and $u=v$ if you see it from $v$-axis}\\
&=\int_{u=0}^{v}\frac1{2v}\ du\\
&=\left.\frac u{2v}\right|_{u=0}^{v}\\
&=\frac12\;;\quad\text{ for } 0\le v\le1,
\end{align}
$$
and
$$
\begin{align}
f_V(v)&=\int_{u=0}^{\frac1v}f_{U,V}(u,v)\ du\ ;\quad\text{the region is bounded by $u=0$ and $u=\frac1v$ if you see it from $v$-axis}\\
&=\int_{u=0}^{\frac1v}\frac1{2v}\ du\\
&=\left.\frac u{2v}\right|_{u=0}^{\frac1v}\\
&=\frac 1{2v^2}\;;\quad\text{ for } 1\le v\le\infty.
\end{align}
$$
It can also be written as
$$
f_V(v)=
\left\{ 
\begin{array}{l l}
\frac12&\;;\quad\text{ for } 0\le v\le1,\\\\
\frac 1{2v^2}&\;;\quad\text{ for } 1\le v\le\infty.
  \end{array} \right.
$$
