Calculate the flux of the vector field $\vec{F}(x,y,z) = 3 \vec{i} + 4 \vec{j} + z \vec{k}$ through the closed circular cylinder of radius $3$ centered about the z-axis for $-6 \leq z \leq 6$, oriented away from the z-axis. Note: a closed cylinder has a top and a bottom.

Flux = $\displaystyle \iint\limits_S \vec{F} \cdot d\vec{A}$ =