Answer
$\dfrac{3a}{4}$
Work Step by Step
The average distance of a point from the origin can be calculated as: $\dfrac{\iiint_{E} \rho dV}{\iiint_{E} dV}$
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 $ or, $=\int_{0}^{\pi} \sin \phi d\phi \int_{0}^{2\pi} d\theta \int_{0}^{a} \rho^3 d\rho$
or, $=[-\cos \phi ]_{0}^{\pi} \times [\theta ]_{0}^{2\pi} [\rho^4/4]_0^a$
or, $=(2) (2 \pi) [\dfrac{a^4}{4}] $
or, $=\pi a^4 $
This the volume of the sphere; $\iiint_{E} dV= \dfrac{4}{3} \pi a^3$
Therefore, the average distance of a point from the origin will become: $\dfrac{\pi a^4 }{\dfrac{4}{3} \pi a^3}=\dfrac{3a}{4}$
