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

There is the problem that these three planes don't enclose a volume, in fact 3 planes in 3d never will, just like how 2 lines in 2d will never enclose an area. — Ninad Munshi, Dec 07 '20 at 05:48

David G. Stork · Answer 1 · 2020-12-09T02:47:35.727

0

Multiply out the polynomial and integrate each term to find:

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

edited Dec 09 '20 at 02:47

answered Dec 07 '20 at 05:50

David G. Stork

should be a comment. please refrain from one-liner answers in the future. – bfff Dec 09 '20 at 02:35
It isn't a comment. It shows the solution. – David G. Stork Dec 09 '20 at 02:46

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