[tex]\vec F(x,y,z)=\langle z,y,x\rangle[/tex]
has divergence 1, so by the divergence theorem, the flux of [tex]\vec F[/tex] across the boundary of [tex]E[/tex] is exactly the volume of [tex]E[/tex],
[tex]\displaystyle\iiint_E\mathrm dV=\int_0^\pi\int_0^{2\pi}\int_0^8\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
or simply [tex]\dfrac43\pi8^3=\boxed{\frac{2048\pi}3}[/tex].
