Let $g$ be the function given by $g ( x ) = \frac { 1 } { \sqrt { x } }$. (a) Find the average value of $g$ on the closed interval $[ 1,4 ]$. (b) Let $S$ be the solid generated when the region bounded by the graph of $y = g ( x )$, the vertical lines $x = 1$ and $x = 4$, and the $x$-axis is revolved about the $x$-axis. Find the volume of $S$. (c) For the solid $S$, given in part (b), find the average value of the areas of the cross sections perpendicular to the $x$-axis. (d) The average value of a function $f$ on the unbounded interval $[ a , \infty )$ is defined to be $\lim _ { b \rightarrow \infty } \left[ \frac { \int _ { a } ^ { b } f ( x ) d x } { b - a } \right]$. Show that the improper integral $\int _ { 4 } ^ { \infty } g ( x ) d x$ is divergent, but the average value of $g$ on the interval $[ 4 , \infty )$ is finite.
Let $g$ be the function given by $g ( x ) = \frac { 1 } { \sqrt { x } }$.\\
(a) Find the average value of $g$ on the closed interval $[ 1,4 ]$.\\
(b) Let $S$ be the solid generated when the region bounded by the graph of $y = g ( x )$, the vertical lines $x = 1$ and $x = 4$, and the $x$-axis is revolved about the $x$-axis. Find the volume of $S$.\\
(c) For the solid $S$, given in part (b), find the average value of the areas of the cross sections perpendicular to the $x$-axis.\\
(d) The average value of a function $f$ on the unbounded interval $[ a , \infty )$ is defined to be $\lim _ { b \rightarrow \infty } \left[ \frac { \int _ { a } ^ { b } f ( x ) d x } { b - a } \right]$. Show that the improper integral $\int _ { 4 } ^ { \infty } g ( x ) d x$ is divergent, but the average value of $g$ on the interval $[ 4 , \infty )$ is finite.