change order of integration Often when I do integrations I need to change the order of integration and this is time consuming. Consider for example \begin{align} \int_0^1\int_0^x f(x,r) \mbox{d}r\mbox{d}x=\int_0^1\int_r^1 f(x,r) \mbox{d}x\mbox{d}r \end{align} and to come up with that I draw a little graph and change the order the usual way. But if I have a quadrupole then things get messy and time consuming. I am hence after some technique to cut down on number of graphs I have to draw. In particular, how would you approach the following change of variable problem in a fast way? $$\int_0^1\int_0^1\int_0^r\int_0^xf(x,r,\alpha,\beta)\mbox{d}\beta \mbox{d}\alpha \mbox{d}r\mbox{d}x=\int_0^1\int_?^?\int_?^?\int_?^?\mbox{d}r\mbox{d}x \mbox{d}\beta \mbox{d}\alpha$$ Many thanks in advance.

First write explicitly the integral domain:
$$0\le x,r\le 1, 0\le \alpha\le r, 0\le \beta\le x.$$ For fixed $\alpha\in[0,1]$, we have $0\le\beta\le 1$. Then we have $\beta\le x\le 1$, and $\alpha\le r\le 1$. Hence, $$\int_0^1\int_0^1\int_0^r\int_0^xf(x,r,\alpha,\beta)\mbox{d}\beta \mbox{d}\alpha \mbox{d}r\mbox{d}x=\int_0^1\int_0^1\int_\beta^1\int_\alpha^1f(x,r,\alpha,\beta)\mbox{d}r\mbox{d}x \mbox{d}\beta \mbox{d}\alpha.$$