# Question about the symmetry of integrals

1. Feb 6, 2014

### LateToTheParty

O.K. , this question is inspired by a physics class I'm taking where we're working out the expectation values of wave functions, but I think the question really belongs in the math section. Thank you in advance for any help. Here goes nothing...
We have a function ψ(x,y,z) = x e$\sqrt{}x2 + y2+ z2$
Now I want to integrate this over all space. So I switch to spherical polar include the Jacobian and I end up with a separable integral of sinθ cubed form 0 to pi which is zero. No problem. But here's where the question pops up. I go to calculate the same function except this time with a z in front of the exponential, and I get a different answer, because I no longer get the sin^3 when I switch to spherical. It just seems really odd to me. I apologize if my formatting leaves something to be desired I'm new to the site, and any help would be greatly appreciated. Any insight on why a arbitrary change would affect the behavior of the integral?

2. Feb 6, 2014

### ShayanJ

In the first case(x in front),the integral is:
$\int_0^\infty \int_0^{2\pi} \int_0^\pi r\sin{\theta}\cos{\varphi} e^r r^2\sin{\theta}d\theta d\varphi dr=\int_0^\infty r^3 e^r dr \int_0^{2\pi} \cos\varphi d\varphi \int_0^\pi \sin^2{\theta} d\theta$
The $\varphi$ integral is equal to zero so the answer is zero.
In the second case(z in front),the integral is:
$\int_0^\infty \int_0^{2\pi} \int_0^\pi r\cos{\theta} e^r r^2\sin{\theta}d\theta d\varphi dr= \int_0^\infty r^3 e^r dr \int_0^{2\pi} d\varphi \int_0^\pi \cos{\theta}\sin{\theta}d\theta$
Now the $\theta$ integral is zero and so again the answer is zero.