Answer
$$\dfrac{31\pi}{6}$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$ Volume=\int^{2\pi}_0 \int^{\pi/2}_0 \int^2_ {cos\phi} p^2 \sin \phi \space dp \space d\phi \space d\theta \\=\dfrac{1}{3} \times \int^{2\pi}_0 \int^{\pi/2}_0 (8-\cos^3\phi) \sin\phi \space d\phi \space d\theta \\=\dfrac{1}{3}\int^{2\pi}_0 [-8 \cos(\phi)+\dfrac{\cos^4(\phi)}{4}]^{\pi/2}_0 \space d\theta \\=\dfrac{1}{3}\int^{2\pi}_0 (8-\dfrac{1}{4})\space d\theta \\=\dfrac{31\pi}{6}$$
