$4\mathrm{.}7.$ When we derived the integral equations for a flat$-$plate boundary layer, the outer boundary of our control volume was a streamline outside the boundary layer $($see Fig. $4\mathrm{.}17).$ Let us now apply the integral equations to a rectangular control volume to calculate the sectional drag coefficient for incompressible flow past a flat plate of length L$.$ Thus, as shown in Fig. P$4\mathrm{.}7,$ the outer boundary is a line parallel to the wall and outside the boundary layer at all x stations. Owing to the growth of the boundary layer, fluid flows through the upper boundary with a velocity $v_e$ which is a function of x $.$ How does the resultant expression compare with equation $(4\mathrm{.}70)?$

Figure P$4\mathrm{.}7$