Changing the bounds of an integral

Question

I am trying to understand the following equality:

$\int_{-\pi}^{\pi}f(y)g(x-y)dy = \int_{x+\pi} ^{x-\pi} f(x-u)g(u)(-du)$

With the substitution $u=x-y$

I understand why $f(y) \rightarrow f(x-u)$ and why $g(x-y)\rightarrow g(u)$ and why $dx\rightarrow -du$.

What I don't understand is why $\pi \rightarrow x-\pi$ and why $-\pi \rightarrow x+ \pi$

Answer 1

When $y=-\pi$, then $u=x-y=x+\pi$, and when $y=\pi$ then $u=x-y=x-\pi$.

Answer 2

Simply substitute $y=\pm \pi$ in $u=x-y$: