Find area bounded by $y=\frac 3x, y=\frac 5x, y=3x, y=6x$ Let S be the area of the region bounded by the curves 
$$y=\frac 3x,\>\>\> y=\frac 5x,\>\>\> y=3x,\>\>\> y=6x$$
Need to find $S$.
The coordinates of the vertices of the resulting figure were found. The problem with the transition from a double integral to a repeated one.
There's another idea. Make a replacement
\begin{cases}
\xi=xy \\ \eta=\frac{y}{x}
\end{cases}
Вut none led to the correct answer.

In polar coordinates, the boundaries are
$$r^2=\frac3{\sin\theta\cos\theta},\>\>\>\>\>r^2=\frac5{\sin\theta\cos\theta},
\>\>\>\>\>\tan\theta = 3,\>\>\>\>\>\tan\theta = 6$$
Thus, the area integral is,
$$S=\int_{\theta_1}^{\theta_2}d\theta \int_{r_1}^{r_2}rdr
=\int_{\theta_1}^{\theta_2}\frac{d\theta}{\sin\theta\cos\theta}=\ln(\tan\theta)|_{\tan^{-1}3}^{\tan^{-1}6}=\ln2$$
