Integral over domain enclosed by curves

Question

Integrate the following function over the domain $D$ which is limited by the curves $xy=1, xy=2, y-x=1, y-x=3.$ \begin{equation} \int\int_{D} (x+y)dxdy \end{equation}

I have drawn out the domain and I have found all the points of intersection between the curves. However, it seems as though maybe a variable change would make this problem easier and simplify the integration bounds.

Any suggestions on how to proceed?

Answer 1

let $u=xy$ and $v=y-x$, we have \begin{align} & \frac{\partial (u,v)}{\partial (x,y)}=\left| \begin{matrix} y & x \\ -1 & 1 \\ \end{matrix} \right|=x+y\Rightarrow J=\frac{1}{x+y} \\ & \int_{1}^{2}{\int_{1}^{3}{dvdu}}=6 \\ \end{align}

Answer 2

For a sensible variable change I would suggest using $xy$ as one variable and then you can probably work it out.