- #1

hm8

- 16

- 0

## Homework Statement

Show that

[itex]\int\int\int \sqrt{x^{2}+y^{2}+z^{2}} [/itex] [itex] e^{-({x^{2}+y^{2}+z^{2}})} dxdydz = \pi/4[/itex] where the bounds of x, y, and z are 0 to infinity

(The improper integral is defined as the limit of a triple integral over the piece of a solid sphere which lies in the first octant as the radius of the sphere increases indefinitely).

## Homework Equations

In spherical coordinates, ρ

^{2}= x

^{2}+ y

^{2}+ z

^{2}

dxdydz = ρ

^{2}sin∅ drho dpho dtheta

## The Attempt at a Solution

I tried converting to spherical coordinates, which gave me

[itex]\int\int\int ρ^{3} [/itex] [itex] e^{-ρ^{2}} sin\phi d\rho d\phi d\theta[/itex]

But I'm not sure what my bounds would be (isn't ρ in relation to theta or phi somehow?) or even if I did, I'm not sure I could integrate it...