Rewrite Integral Using Transformation

Question

I'm trying to do a transformation of an integration. I have the integral $$\int_{0}^{\pi}\int_{0}^{\pi-y}(x+y)\cos(x-y)dxdy$$

and the transformation $(u,v)=(x+y,-x+y)$. I got the integral $$2\iint _{D}u\cos(-v)dvdu$$ but I don't know how to calculate the new region $D$ for $u$ and $v$. Can anyone help me please?

Answer 1

Draw a picture. Your region is triangular and the transformation is linear. If you can figure out where the three vertices of the triangle get mapped, you can sketch the new region and go from there.

Answer 2

From $$ \begin{align} 0&\le y\le \pi\\ 0&\le x\le \pi-y \end{align} $$

you find

$$ \begin{align} 0&\le x+y\le \pi\\ -\pi&\le -x+y\le \pi \end{align} $$