Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = \ln x$. (a) Find the area of the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$. (b) Find the volume of the solid generated when the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$ is revolved about the line $y = 4$. (c) Let $h$ be the function given by $h ( x ) = f ( x ) - g ( x )$. Find the absolute minimum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$, and find the absolute maximum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$. Show the analysis that leads to your answers.
Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = \ln x$.\\
(a) Find the area of the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$.\\
(b) Find the volume of the solid generated when the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$ is revolved about the line $y = 4$.\\
(c) Let $h$ be the function given by $h ( x ) = f ( x ) - g ( x )$. Find the absolute minimum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$, and find the absolute maximum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$. Show the analysis that leads to your answers.