I'm currently trying to understand a single passage in a proof, which contains the following identity: $$\int_a^b\varphi'(x)\int_a^xg(t)\,dt\,dx=\int_a^bg(t)\int_t^b\varphi'(x)\,dx\,dt,$$ where $[a,b]\subset\mathbb{R}$, $\varphi\in\mathcal{C}^\infty_c([a,b])$ and $g\in L^1_\text{loc}([a,b])$.
The Fubini-Tonelli Theorem has been applied, but I don't understand why the bounds are changing from $[a,x]$ to $[t,b]$. Any clues?
In the first integral, $x$ can take any value from $a$ to $B$ and, for each such $x$, $t$ can take any value from $a$ to $x$. Therefore, the set of all pairs $(x,t)$ is the triangle whose vertices are $(a,a)$, $(b,a)$, and $(b,b)$.
In the second integral, $t$ can take any value from $a$ to $b$ and, for each such $t$, $x$ can take any value from $t$ to $b$. But this gives you again the triangle whose vertices are $(a,a)$, $(b,a)$, and $(b,b)$.
So, the limits of the integral of the RHS are chosen so that they match the integral of the LHS.