Solve the following integral $\iiint \left(x+y+z\right)\left(x-y-z\right)\left(x+y-z\right)dV$

Question

Hello there I am looking how to solve this integral

$$\iiint \left(x+y+z\right)\left(x-y-z\right)\left(x+y-z\right)dV$$

using change of variable under the following planes:

$x\:+\:y\:+\:z\:=\:0$

$x\:+\:y\:-\:z\:=\:0$

$x\:-\:y\:-\:z\:=\:0$

Please give some advices to do solve it or any hints you have in mind...

Thanks

Answer 1

Multiply out the polynomial and integrate each term to find:

$$\frac{1}{6} x y z \left(2 x^2-2 y^2-3 y z-2 z^2\right)$$