Answer
$$\pi^2$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$\int^{\pi}_0 \int^{\pi}_0 \int^{2sin \phi}_0 p^2 \space \sin\phi \space dp \space d\phi \space d\theta =\dfrac{8}{3} \times \int^{\pi}_0\int^{\pi}_0sin^4\phi d\phi \space d\theta \\=\dfrac{8}{3} \times \int^{\pi}_0 ([-\dfrac{\sin^3\phi \times \cos\phi}{4}]^\pi_0+\dfrac{3}{4} \times \int^{\pi}_0 \sin^2 \phi ) \space d \phi \\=2 \times \int^{\pi}_0 \int^{\pi}_0 \sin^2 \phi d\phi d\theta \\=\int^{\pi}_0 [\theta-\dfrac{ \sin (2\theta)}{2}]^\pi_0 d\theta \\=\int^{\pi}_0 \pi d\theta \\=\pi^2$$
