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