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## 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?