This might be considered as too general of a question but it has been bugging me for quite a while now: How does one change integration domains after using the Fubini-Tonelli's Theorem? Especially in higher dimensions when things are not "drawable"? Are there any algebraic ways or systematic ways to figure out the integration domains? Here I give one example of such situation just to illustrate what I meant:
To me, it is not immediately clear how we changed the integration domain like such. It makes sense somewhat that we have $\| \mathbf{x}' - \mathbf{y}' \| \leq x_N$ as $\mathbf{y}' \in B_{N-1}(\mathbf{x}', x_N)$, but how exactly would one start to think of such change of integration domain?
Tonelli's theorem only works when the domains of integration on each integral are constant. Use indicator functions to convert any seemingly non-constant domains of integration into constant ones. In your problem, you can convert $B_{N - 1}(x', x_N)$ into the whole space and multiply the integrand by $I(y' \in B_{N - 1}(x', x_N))$. Now you apply Tonelli/Fubini to interchange integrals.