Why this triple integral equals zero?

[topito2]

Hello everyone, I have the this inquiry:

if I compute de following integral:
http://micurso.orgfree.com/Picture1.jpg
by numerical methods I get cero as a result. I used Maxima and Mathematica and their functions for numerical integration give me an answer equal to cero.
But, if I apply transformation with spherical coordinates the result turns out to be equal to pi^2 / 40.
I can't seem to find an explanation for this. Could any of you guys give me a hand with this? Thank you in advance.

 

[jedishrfu]

Well we'd need to see how you transformed the integral and its limits into spherical coordinates before we could even attempt to say what happened.

One question I had was whether you transformed the dxdydz properly or not.

https://en.m.wikipedia.org/wiki/Volume_element

Where you can see that dXdydz transforms into:

$dV = \rho^2 sin \theta d\rho d\theta d\phi$

 

[DrClaude]

The thread title asks why this is zero, which is quite easy to show. The first two integrations, over z and y, are trivial to do, and the result is an odd function of x.

 

[topito2]

Thank you so much for your answers, I rechecked what I did this morning and I found out what happens when calculating the integral in spherical coordinates.

Again, thank you so much for your time and cooperation, you guys rock!