# A triple integration that need change of boundary

Evaluate Integral: (Without calculator, Just by hand)

$$\int_0^1\int_0^{1-x}\int_y^1\frac{\sin(\pi z)}{z(z-2)}\,dz\,dy\,dx$$

The answer of the problem uses a visualization of the boundary and change the order of integration such that $$0\leq z\leq 1; ~~ 0\leq y\leq z ; ~~ 0\leq x\leq 1-y$$ and changed the order to $$dx\,dy\,dz$$. And then it solves easily. I want to know is there any way to solve this change of boundaries without visualizing the shape? and by using only algebraic inequalities? How?

Note that I don't want to use this visualization. because sometimes for some planes that are not very well visualized, the answer gets a lot complicated. and I want a method to find a way to change order of integration. At least in problems involving intersection of planes and lines.

• I know that visualizing is the best way. But sometime plotting them is very difficult by hand for example for not very intuitive planes like $2x + 3y + 4z = 5$ , $x + y + 3z = 2$ , $-x+y+4z = 15$ (Just random numbers, Not sure what they actually make). And you must answer these question in only 15 minutes without any calculator or anything else. Commented Jun 25, 2019 at 22:09