Evaluate $\int_S \vec F\cdot d\vec A$, where $\vec F =(bx/a)\vec i +(ay/b)\vec j$ and $S$ is the elliptic cylinder oriented away from the $z$-axis, and given by $x^2/a^2 +y^2/b^2=1$, $|z|\leq c$, where $a$, $b$, $c$ are positive constants.

$\int_S \vec F\cdot d\vec A =$