Prove the improper triple integral equals 2*Pi

[alanthreonus]

Homework Statement

Show that

$$\int^{infinity}_{-infinity}\int^{infinity}_{-infinity}\int^{infinity}_{-infinity}sqrt(x^2+y^2+z^2)e^-^(^x^2^+^y^2^+^z^2^)dxdydz = 2\pi$$

Homework Equations

$$x^2+y^2+z^2 = \rho^2$$

The Attempt at a Solution

I converted to spherical coordinates to get

$$\int^{2\pi}_{0}\int^{2\pi}_{0}\int^{infinity}_{0}\rho^3e^-^\rho^2sin\phi d\rho d\phi d\theta$$

But I don't think I can integrate that. Am I approaching this the right way?

 

[Jerbearrrrrr]

The bounds are independent.
You can get rid of the (phi) and (theta) integrals (ie factor them out).

As for the remaining integrand, you have a power of (rho) multiplied by something you can integrate (may take a second thought to get this 'something' right).
What method does this suggest?

Spoiler

Consider:
(int) x² . x exp(-x²) dx

 

[alanthreonus]

OK, integration by parts gives

(-1/2e^(-rho^2))(rho^2+1)

Which is evaluated from 0 to infinity, but unless I'm missing something, the limit at infinity is undefined.

And something else occurred to me. If I integrate with respect to phi, I'll get -cos(phi), which evaluated from 0 to 2*pi is 0. So does that mean I messed up the conversion to spherical coordinates?

 

[guerom00]

Kinda… In spherical coordinates, what you call phi goes from 0 to Pi, not 2*pi :)
As for the integration over rho, it definitely converges as rho^3 Exp[-rho^2] is 0 at 0 and goes to 0 at infinity (the exponential is stronger than any power of rho) :)

 

[vela]

OK, integration by parts gives

(-1/2e^(-rho^2))(rho^2+1)

Which is evaluated from 0 to infinity, but unless I'm missing something, the limit at infinity is undefined.

You can put the exponential in the denominator and use L'Hopital's rule to evaluate the limit at infinity.

 

[alanthreonus]

Thanks a bunch guys, I got it now. And now I feel dumb for not seeing all these things I missed.

 

[Planefreak]

I'm kind of lost. I did all the work up to the very end and have just the limit function left but can't get it to work out.

Can someone check this and let me know what else I need to do or what I messed up on?

I have:

2$$\pi$$ * lim (t $$\rightarrow$$ $$\infty$$ [ -e^{-t^2}/2 * (t² + 1) + 1/2 ]

I believe that the way I have this written down it is equal to some undefined number because we find that the exponential part goes to 0 while t² becomes $$\infty$$

Thanks in advance, Travis

 

[vela]

$0\cdot\infty$ is an indeterminate form. You can use L'Hopital's rule to evaluate the limit.

 

[Planefreak]

Now I really feel dumb... like the person who started the thread I messed up my limits. For some reason I thought phi was [0, pi/2] so when I finished everything I came up with just pi as a result. Once I fixed this it worked out to 2*pi