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