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.


1 Answer 1


In my experience, changing the order of integration when the limits are variable is sometimes possible, but never intuitive. Plotting the region is one of those kinds of steps that I can't skip without getting something wrong, so I simply don't skip that step. Ever. It is, admittedly, even harder in 3D than in 2D, but I would still recommend plotting the region and visualizing each variable as functions of other variables. That may not be the answer you want to hear, but it's my answer. Naturally, if the limits are all constants, then it's much easier to change the order - your region is rectangular.

  • $\begingroup$ 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. $\endgroup$
    – amir na
    Commented Jun 25, 2019 at 22:09
  • 1
    $\begingroup$ I would use traces to plot planes like those quickly: set variables equal to zero and see what happens with the rest of the equation. $\endgroup$ Commented Jun 25, 2019 at 22:18

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