Answer
$\dfrac{3a}{4}$
Work Step by Step
The average distance of a point from the origin is given by: $\dfrac{\iiint_{E} \rho dV}{\iiint_{E} dV}$
Now, $\iiint_{E} \rho dV =\int_{0}^{\pi} \int_{0}^{2\pi}\int_{0}^{a} \rho( \rho^2 \sin \phi) \rho d\rho d\theta d\phi \\=\int_{0}^{\pi} \sin \phi d\phi \int_{0}^{2\pi} d\theta \int_{0}^{a} \rho^3 d\rho \\=[-\cos \phi ]_{0}^{\pi} \times [\theta ]_{0}^{2\pi} [\rho^4/4]_0^a \\=(2) (2 \pi) [\dfrac{a^4}{4}] \\=\pi a^4 $
and $\iiint_{E} dV$ is the volume of the sphere, and is equal to $ \dfrac{4}{3} \pi a^3$
Thus, the average distance of a point from the origin is given by: $\dfrac{\pi a^4 }{\dfrac{4}{3} \pi a^3}=\dfrac{3a}{4}$
