Average of a quantity in all directions

[matteo86bo]

Homework Statement

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

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

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
\langle A\rangle_r=\left\langle n \right\rangle R
so, where this factor 4/3 comes from?

 

[tjackson3]

Well, the way you typically find the average value of some function f(x) on a domain \Omega is the following:

<f> = \frac{\int_{\Omega}\ f(x)\ d\Omega}{\int_{\Omega}\ d\Omega

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