Problem

The equation of a curve is $y=4x^{\frac{1}{2}}-x$. The curve has a maximum point when $x=a$ and crosses the $x$-axis at the point with coordinates $(b,0)$, where $b>0$. The shaded region is bounded by the curve, the line $x=a$ and the $x$-axis (see diagram).

(a) Find the value of $a$.

(b) Find the exact area of the shaded region.