Integrating a function in infinite space / sphere

[brollysan]

Homework Statement

Integrate exp(-3(sqrt(x**2 + z**2 + y**2))) over infinite space [-inf, inf] on xyz

Well transforming to spherical coordinates leaves me with the equation at 3.attempts at a..()
but here is my problem, how can you equate an integral over an infinite space to a spherical integral? An infinitely big sphere has at most an infinite diameter, where is the guarantee that your function doesn't land outside the sphere? Am I not getting something fundamental or does this seem like a paradox? A sphere inside a cube will never contain all the points of the cube, does the transformation of coordinates warp all points of xyz into a sphere? Then we have the problem of sphere inside the cube again..

The limits of the new integral will be? 0,2pi for the angels and 0, inf for p? But how can that be right, a point from the centre of the sphere to its surface has length of r which is d/2 = inf/2 ??

Homework Equations

The Attempt at a Solution

Arrived at 3x- integral: (p**2)(exp(-3p))sin(phi)dpdphidtheta but not sure where the limits should go

 

[Dick]

Arrived at 3x- integral: (p**2)(exp(-3p))sin(phi)dpdphidtheta but not sure where the limits should go

The integral part looks ok. I'm not sure what the '3x-' part is supposed to be. And sure, you are integrating over ALL space. So the limits are the total range of each spherical coordinate. Which means both angles don't go 0 to 2pi. Only one does. Review spherical coordinates if you are unsure. And I would stop fretting about big spheres. As I said, you are integrating over all of space. Any 'sphere' will fit in there, if that makes you feel better.