How do I evaluate$ \iint_D (x+2y)^2 e^{x-y}dxdy$ , where $D$ is the region in the $x-y$ plane bounded by a parallelogram?

Question

The vertices of the parallelogram are $(0,0), (1,1), (2,-1)$ and $(3,0)$ I was integrating the equation for $y = x$, and $y= \frac12x + 3$ and I realized this might need a "change of variables".
So how would I go about doing that?

Answer 1

Suppose we let $A=(0,0)$, $B=(1,1)$, $C=(2,-1)$ and $D=(3,0)$. Then, the equations of the lines of the parallelogram are $\overline{AB}:x-y=0$, $\overline{AC}:x+2y=0$, $\overline{BD}:x+2y=3$ and $\overline{CD}:x-y=3$. This suggests making the change of variable $u=x-y$ and $v=x+2y$ (so that the bounds of the resulting double integral are constant). Can you continue?

You might find the following inverse transform useful for computing the Jacobian: $$x=\frac{2u+v}{3},\quad y=\frac{v-u}{3}$$