I am currently working on a calculus problem and I'm having trouble finding a suitable variable substitution. I would greatly appreciate any help or guidance you can provide.
The problem I am trying to solve is as follows:
Calculate the integral $\iint_D \frac{x+y}{\sqrt{2 x-y}} d x d y$ using an appropriate variable substitution, where $D$ represents the parallelogram with vertices at $(1,1),(2,0),(1,-2)$ and $(0,-1).$
Here's what I have attempted so far:
I understand that using a suitable variable substitution can simplify the integral and make it easier to evaluate. However, I'm unsure about how to choose the right substitution in this case. I have tried different approaches, but none of them seemed to lead me closer to a solution. Could someone please guide me on how to determine a suitable variable substitution for this problem? Any explanation or step-by-step approach would be immensely helpful.
Let $A(1,1), B(2,0), C(1,-2)$ and $D(0,-1)$, then we have $AB//CD, AD//BC$
Define: $$u=x+y,~~v=2x-y$$
then the original region $D$ is converted to
$$(u,v)\in [-1,2]\times[-1,4]$$
Next, you need to compute Jacobian and do the integral with respect to $u$ and $v$. Namely,
$$\int_{-1}^4\int_{-1}^2 \frac{u}{\sqrt v}|J|dudv$$
Can you proceed from here?