Problem

The diagram shows the curve with equation $y=\frac{1}{2}\sqrt{x}$ and the point $P$ with coordinates $\left(9,\frac{3}{2}\right)$. The shaded region is bounded by the curve and the lines $x=0$ and $y=\frac{3}{2}$.

(a) Find the area of the shaded region.

(b) The shaded region is rotated through $360^\circ$ about the $y$-axis.

Find the exact volume of the solid produced.