$2.$ Let $\backslash ($ R $\backslash )$ be the region enclosed by the graph of $\backslash ($ f$($x$)=-$x$^\{4\}+2\mathrm{.}3$ x$^\{3\}+5\backslash )$ and the horizontal line $\backslash ($ y$=5\backslash ),$ as shown in the figure above.

$($a$)$ Find the volume of the solid generated when $\backslash (\backslash$boldsymbol$\{$R$\}\backslash )$ is rotated about the line $\backslash ($ y$=-3\backslash ).$

$($b$)$ Region $\backslash (\backslash$boldsymbol$\{$R$\}\backslash )$ is the base of a solid. For this solid, each cross section perpendicular to the $-\backslash ($ x $\backslash )-$axis is a rectangle whose height is twice as long as the length in $\backslash ($ R $\backslash ).$ Find the volume of the solid.

$($c$)$ The vertical line $\backslash ($ x$=$h $\backslash )$ divides region $\backslash ($ R $\backslash )$ into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value $\backslash ($ h $\backslash ).$