Let $f$ and $g$ be the functions given by $f ( x ) = 2 x ( 1 - x )$ and $g ( x ) = 3 ( x - 1 ) \sqrt { x }$ for $0 \leq x \leq 1$. The graphs of $f$ and $g$ are shown in the figure above. (a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$. (b) Find the volume of the solid generated when the shaded region enclosed by the graphs of $f$ and $g$ is revolved about the horizontal line $y = 2$. (c) Let $h$ be the function given by $h ( x ) = k x ( 1 - x )$ for $0 \leq x \leq 1$. For each $k > 0$, the region (not shown) enclosed by the graphs of $h$ and $g$ is the base of a solid with square cross sections perpendicular to the $x$-axis. There is a value of $k$ for which the volume of this solid is equal to 15 . Write, but do not solve, an equation involving an integral expression that could be used to find the value of $k$.
: \frac { d ^ { 2 } y } { d x ^ { 2 } }
Let $f$ and $g$ be the functions given by $f ( x ) = 2 x ( 1 - x )$ and $g ( x ) = 3 ( x - 1 ) \sqrt { x }$ for $0 \leq x \leq 1$. The graphs of $f$ and $g$ are shown in the figure above.
(a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$.
(b) Find the volume of the solid generated when the shaded region enclosed by the graphs of $f$ and $g$ is revolved about the horizontal line $y = 2$.
(c) Let $h$ be the function given by $h ( x ) = k x ( 1 - x )$ for $0 \leq x \leq 1$. For each $k > 0$, the region (not shown) enclosed by the graphs of $h$ and $g$ is the base of a solid with square cross sections perpendicular to the $x$-axis. There is a value of $k$ for which the volume of this solid is equal to 15 . Write, but do not solve, an equation involving an integral expression that could be used to find the value of $k$.