# How to find a suitable substitution? $\iint_D \frac{x+y}{\sqrt{2 x-y}} d x d y$

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?

• media.discordapp.net/attachments/1108131788229971998/… How did you arrive at that u and v? A friend of mine demonstrated the use of base change matrices, which can be helpful for these types of substitutions. Commented May 30, 2023 at 15:18
• Never mind, I get it now. It's really clever to view it as parallel lines. Commented May 30, 2023 at 15:19
• Yes, use parallel line, so it is just a translation. Commented May 30, 2023 at 15:23