Problem $1(20$ points$)$: An incompressible fluid flows past a solid $($impermeable$)$ flat plate as shown below. At the inlet on the left$-$hand side of the plate, the flow is uniform with velocity magnitude $U_0.$ At the outlet on the right$-$hand side of the plate, the flow is parabolic with a velocity profile given by $u(y)=U_0(2\frac{y}{}-\frac{y^2}{{}^2}).$

Compute the volume flow rate $Q$ across the top surface of the control volume. Express your answer in terms of $U_0,b,$ and $.$