$(5$ points$)$ Write the surface integral ${}_{S^2}$vec$(F)*dvec(S),$ as a double integral, where vec$(F)=($:$x^{2}y,0,z$:$)$ and $S$ is the paraboloid $z=4-x^2-y^2$ above the $xy-$plane.

$($Hint: A parametrization of $S$ is vec$(r)(u,v)=,0u2,0v2.)$