Answer
$ \dfrac{4096 \pi}{21}$
Work Step by Step
The spherical coordinates system can be expressed as:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta; z=\rho \cos \phi$ and $\rho=\sqrt {x^2+y^2+z^2}$
or, $ \rho^2=x^2+y^2+z^2$
We define the jacobian for spherical coordinates as:
$\phi^2 \sin \phi$,
Therefore, $\int_0^{2 \pi} \int_0^{\pi/2} \int_0^{4 \cos \phi} \rho^3 i \rho^2 \rho^2 \sin \phi d\rho d \phi d \theta=\int_0^{2 \pi} \int_0^{\pi/2} [ \rho^6/6]_0^{4 \cos \phi} \sin \phi d\rho d \phi d \ \theta$
and consider $ u=\cos \phi$
Thus, we have:
$E=\dfrac{(4)^6}{6} [2 \pi \times(\dfrac{-\cos^7 \pi}{7})]_0^{\pi/2} \\ = \dfrac{4096 \pi}{21}$
