Answer
$$\dfrac{ 3a}{4}$$
Work Step by Step
$I_{avg}=\dfrac{3}{4 \pi a^3} \int_0^{2\pi} \int_{0}^{\pi} \int_{0}^{a} \rho^3 \sin \phi d \ rho d \theta \\=\dfrac{3a}{16 \pi} \int_0^{2\pi} \int_{0}^{\pi} \sin \phi \ d \phi \ d\theta \\=\int_0^{2\pi} \dfrac{3a d\theta }{8\pi} \\= \dfrac{3a}{8\pi} \times (2\pi-0) \\=\dfrac{ 3a}{4}$
