Answer
$$\dfrac{8\pi}{3}$$
Work Step by Step
Our aim is to integrate the integral as follows:
$ Volume=\int^{2\pi}_0 \int^\pi_0 \int^{1-cos\phi}_0 p^2\sin\phi \space dp \space d\phi \space d\theta $
or, $=\dfrac{1}{3}\int^{2\pi}_0 \int^\pi_0 (1-\cos\phi)^3 \sin \phi \space d\phi \space d\theta $
or, $=\dfrac{1}{3}\int^{2\pi}_0 [\dfrac{(1- \cos\phi)^4}{4}]^\pi_0 d\theta $
or, $=\dfrac{1}{12} \times 16 \times \int^{2\pi}_0 d\theta $
so, $ Volume =\dfrac{8\pi}{3}$
