If I have an integral of the form
$$ I = \int_{-T}^T dx\int_{-T}^T dy\ i(x - y) \tag1$$
Where $i$ is any function depending just on the relative variable, i.e., $x - y$. But, let's suppose that I need to perform the following change of variable:
$$ r = \frac{x + y}{2}, \quad R = x - y \tag2$$
Then, I think that $I$ will become
$$ I = \int_{-\sqrt{2}T/2}^{\sqrt{2}T/2}dr \int_{-\sqrt{2}T}^{\sqrt{2}T}dR\ i(R) \tag3$$
But I'm not sure. My reasoning to get this new limits is as follows: from Eq. (2), you can imagine $R$ as the module of a 2D vector in the cartesian plane, so
$$\vec{R}_{maximum} = (T, 0) - (0, T) \Rightarrow R_{maximum} = \sqrt{2}T$$
Similarly, $$R_{minimum} = -R_{maximum}$$
Moreover, $$\vec{r}_{maximum} = 0.5((T, 0) + (0, T)) \Rightarrow r_{maximum} = \frac{\sqrt{2}T}{2}$$
and, as before,
$$r_{minimum} = -r_{maximum}$$
Is this correct or did I fail something?