Answer
$$\dfrac{2\pi a^3}{3}$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$ Volume =\int^{2\pi}_0 \int^{2\pi/3}_{\pi/3} \int^a_0 p^2 \space
\sin\phi \space dp \space d\phi \space d\theta \\=\int^{2\pi}_0 \int^{2\pi/3}_{\pi/3} \dfrac{a^3}{3} \times \sin \phi \space d\phi \space
d\theta \\=\dfrac{a^3}{3}\int^{2\pi}_0[-\cos\phi]^{2\pi/3}_{\pi/3} \space d\theta \\=\dfrac{a^3}{3}\times \int^{2\pi}_0(\dfrac{1}{2}+\dfrac{1}{2}) \space d\theta \\=\dfrac{2\pi a^3}{3}$$
