Answer
$$\dfrac{(4\sqrt{2}-5)}{\sqrt{2}}\pi $$
Work Step by Step
Our aim is to integrate the integral as follows:
$$\int^1_0 \int^{\pi}_0 \int^{\pi/4}_0 12 \space p \sin^3 \phi \space d\phi \space d\theta \space dp =\int^1_0 \int^{\pi}_0 (12 p[\dfrac{-\times \sin^2 \phi \cos \phi}{3}]^{\pi/4}_0+8p \int^{\pi/4}_0 \sin \phi d\phi) \space d\theta \space dp \\=\int^1_0 \int^{\pi}_0 (-\dfrac{2p}{\sqrt{2}}-8p[\cos\phi]^{\pi/4}_0)\space d\theta \space dp \\=\pi \int^1_0 (8p-\dfrac{10p}{\sqrt{2}})\space dp \\=\pi \times [4p^2-\dfrac{5p^2}{\sqrt{2}}]^1_0 \\=\dfrac{(4\sqrt{2}-5)}{\sqrt{2}}\pi $$
