$2.)(10$ pts$)$ Using the transformation $u=\frac{y}{x}$ and $v=xy,$ compute the integral

${}_{R}x{y}^{3}dA$

where $R$ is the region in the first quadrant enclosed by the graphs of the following equations: $y=x,y=3x,xy=1,$ and $xy=4.$

$($You do not need to sketch anything, but you need to display the equations for your transformation, the calculation of the Jacobian, and the calculation of the integral in the new coordinates$).$