Problem $9$: Water flows in a two$-$dimensional channel of width $W$ and depth $D$ and a velocity

profile as shown. $V_s$ is the velocity at the surface midway between the channel walls. Find the

discharge in the channel in terms of $V_s,D,$ and $W.$ See: Plot of Velocity Profile

$V(x,y)=V_s(1-\frac{4x^2}{W^2})(1-\frac{y^2}{D^2})$