Suppose $X_1$ and $X_2$ have the joint pdf
$$f_{X_1,X_2}\left(x_1,x_2 \right)=4x_1 x_2,\quad\text{for}\quad 0<x_1<1, 0<x_2<1$$
From that I found the joint pdf of $Y_1=\frac{X_1}{X_2}$ and $Y_2=X_1 X_2$ to be
$$f_{Y_1,Y_2} \left(y_1,y_2 \right)=2 \frac{y_2}{y_1}\quad \text{for}\quad \left(y_1,y_2 \right) \in \mathcal{T}$$
where $\mathcal{T}=\{ \left(y_1,y_2 \right): y_1,y_2>0,y_2<\frac{1}{y_1},y_2<y_1 \}$
That is, in the $ \left(y_1, y_2 \right)$ plane, the area of the pdf is bounded by the 45 degree line and the hyperbola $\frac{1}{y_1}$. Based on that I can easily derive the marginal of $Y_1$ for the two cases
$0<y_1<1$
$1<y_1<\infty$
But what about the marginal of $Y_2$? $Y_1$ is not bounded above and is bounded below by zero, right? How can I integrate out $Y_1$ if the integral from 0 to infinity diverges? Which begs the question, have I done something wrong in the above?