Problem $6$: $(10$ points$)$ A Bundt cake $S$ can be obtained by taking the region $R$ bounded above by $y=2-\frac{1}{2}(x-3{)}^2$ and below by the $x-$axis, and revolving it around the $y-$axis.

$($a$.)$ Graph the region $R$ and solve $y=2-\frac{1}{2}(x-3{)}^2$ for $x$ to find the equations of the curves that bound $R$ to the right and the left.

$($b$.)$ Find the volume of the cake $S$ using the washer method.