Mean density from a density function $rho(r)$

If you want a mean, you have to normalise by the volume of the region: $$bar{rho}=frac{int^R_0rho(r)r^2;text{d}{r}}{int^R_0r^2;text{d}{r}}.$$ Note you need the volume element in each integral, which is $4pi r^2;text{d}r$ for spherical symmetry.

Note the units too - the numerator has units $rm[density]times[volume]$ and the denominator has units $rm[volume]$, so overall we have units of $rm[density]$, which is good.