We consider the nuclear charge distribution to be uniform i.e., $$\rho(r)=Constant$$
Also, Elmentary volume in spherical-polar co-ordiantes
$$d\tau= r^2 \sin\theta drd\theta d\phi$$
Now, $$\left< r^2 \right> =\frac{\iiint r^2 \rho(r)d\tau}{\iiint \rho(r)d\tau}$$
$$\left< r^2 \right> =\frac{\iiint r^2\rho(r) \times r^2 \sin\theta drd\theta d\phi}{\iiint r^2 \rho(r) \sin\theta drd\theta d\phi}$$
$$r:0\to R$$
$$\theta:0\to \pi$$
$$\phi : 0 \to 2\pi$$
$$\left< r^2 \right> =\frac{\int_0^R r^4 dr}{\int_0^R r^2 dr}$$
$$\left< r^2 \right> =\frac{R^5/5}{R^3/3}=\frac{3R^2}{5}$$
Similarly, $$\left< r \right> =\frac{\iiint r\rho(r) \times r^2 \sin\theta drd\theta d\phi}{\iiint r^2\rho(r) \sin\theta drd\theta d\phi}$$
$$\left< r \right> =\frac{\int_0^R r^3dr}{\int_0^R r^2 dr}=\frac{R^4/4}{R^3/3}=\frac{3R}{4}$$