Let $R$ be the region enclosed by the graph of $f ( x ) = x ^ { 4 } - 2.3 x ^ { 3 } + 4$ and the horizontal line $y = 4$, as shown in the figure above. (a) Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = - 2$. (b) Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with a leg in $R$. Find the volume of the solid. (c) The vertical line $x = k$ divides $R$ into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value $k$.
Let $R$ be the region enclosed by the graph of $f ( x ) = x ^ { 4 } - 2.3 x ^ { 3 } + 4$ and the horizontal line $y = 4$, as shown in the figure above.\\
(a) Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = - 2$.\\
(b) Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with a leg in $R$. Find the volume of the solid.\\
(c) The vertical line $x = k$ divides $R$ into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value $k$.