Problem

The curve with equation $y = 2x - 8x^{\frac{1}{2}}$ has a minimum point at $A$ and intersects the positive $x$-axis at $B$.

(a) Find the coordinates of $A$ and $B$.

(b)

The diagram shows the curve with equation $y = 2x - 8x^{\frac{1}{2}}$ and the line $AB$. It is given that the equation of $AB$ is $y = \frac{2x - 32}{3}$.

Find the area of the shaded region between the curve and the line.