A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by
v = $\langle x - y, z + y + 3, z^2 \rangle$ and the net is described by the equation $y = 1 - x^2 - z^2$, $y \ge 0$, and oriented in the positive y-direction.
(Use symbolic notation and fractions where needed.)
$\iint_S \mathbf{v} \cdot d\mathbf{S} = $
