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