Answer
$$\dfrac{a^3\pi}{18}$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$ Volume =\int^{\pi/6}_0 \int^{\pi/2}_0 \int^a_0 p^2 \sin\phi \space dp \space d\phi \space d\theta \\=\dfrac{a^3}{3} \times \int^{\pi/6}_0 \int^{\pi/2}_0 \sin\phi \space d\phi \space d\theta \\=\dfrac{a^3}{3} \times \int^{\pi/6}_0 \space d\theta \\=\dfrac{a^3\pi}{18}$$
