Let $f$ and $g$ be the functions given by $f ( x ) = 1 + \sin ( 2 x )$ and $g ( x ) = e ^ { x / 2 }$. Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$ as shown in the figure above. (a) Find the area of $R$. (b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis. (c) The region $R$ is the base of a solid. For this solid, the cross sections perpendicular to the $x$-axis are semicircles with diameters extending from $y = f ( x )$ to $y = g ( x )$. Find the volume of this solid.
Correct limits in an integral in (a), (b), or (c).
Let $f$ and $g$ be the functions given by $f ( x ) = 1 + \sin ( 2 x )$ and $g ( x ) = e ^ { x / 2 }$. Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$ as shown in the figure above.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) The region $R$ is the base of a solid. For this solid, the cross sections perpendicular to the $x$-axis are semicircles with diameters extending from $y = f ( x )$ to $y = g ( x )$. Find the volume of this solid.