Average of a quantity in all directions

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  • #1

Homework Statement

I have to compute the average of [itex]A[/itex] in all directions for a sphere of radius [itex]R[/itex].
For the direction [itex]r[/itex], [itex]A[/itex] is defined as:

[itex]A=\int_0^R n(r)dr[/itex]
where [itex]n(r)[/itex] is the density profile of the sphere.
The answer should be
[itex]\langle A\rangle=\frac{4}{3}\left\langle n \right\rangle R[/itex]

2. The attempt at a solution

I don't know how to compute the average over all directions. I mean for direction r the average value of A is simply
[itex]\langle A\rangle_r=\left\langle n \right\rangle R[/itex]
so, where this factor 4/3 comes from?
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  • #2
Well, the way you typically find the average value of some function f(x) on a domain [itex]\Omega[/itex] is the following:

[tex]<f> = \frac{\int_{\Omega}\ f(x)\ d\Omega}{\int_{\Omega}\ d\Omega[/tex]

Note that the quantity in the denominator is just the area of the domain

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