so the answer I have is something like
$\int_{0}^{2}\int_{-v}^{0}...dudv$, I'm ok with the jacobin matrix, but I just don't know how to get the bounds of u, v which are 0 to 2 and -v to 0
It looks like two regions that can be the new region for uv variables, which one should I use?
You have equations $y=x$ and $y=-x+2$ and $y=0,$ which you found to be the equations of the three lines.
Now look at the first equation $y=x.$ This is equivalent to $y-x=0.$ Observe that $u=y-x$ by the given transformation. This implies the line $y-x=0$ transforms to $u=0.$
Look at the second equation $y=-x+2.$ This implies $y+x=2.$ From your given transformation $v=y+x,$ so $y+x=2$ transforms to $v=2.$
Look at the third equation $y=0.$ Observe that $y=\frac{u+v}{2}$ by your given transformation. This means $y=0$ transforms to $\frac{u+v}{2}=0$ or $u+v=0$ multiplying both sides by $2.$
Thus the new triangle in the $u-v$ plane is bounded by equations $u+v=0$ , $v=2$ and $u=0.$ Graph this. Does this help ?