Figures 1 and 2 illustrate regions in the first quadrant associated with the graphs of $y = \frac { 1 } { x }$ and $y = \frac { 1 } { x ^ { 2 } }$, respectively. In Figure 1, let $R$ be the region bounded by the graph of $y = \frac { 1 } { x }$, the $x$-axis, and the vertical lines $x = 1$ and $x = 5$. In Figure 2, let $W$ be the unbounded region between the graph of $y = \frac { 1 } { x ^ { 2 } }$ and the $x$-axis that lies to the right of the vertical line $x = 3$. (a) Find the area of region $R$. (b) Region $R$ is the base of a solid. For the solid, at each $x$ the cross section perpendicular to the $x$-axis is a rectangle with area given by $x e ^ { x / 5 }$. Find the volume of the solid. (c) Find the volume of the solid generated when the unbounded region $W$ is revolved about the $x$-axis.
Figures 1 and 2 illustrate regions in the first quadrant associated with the graphs of $y = \frac { 1 } { x }$ and $y = \frac { 1 } { x ^ { 2 } }$, respectively. In Figure 1, let $R$ be the region bounded by the graph of $y = \frac { 1 } { x }$, the $x$-axis, and the vertical lines $x = 1$ and $x = 5$. In Figure 2, let $W$ be the unbounded region between the graph of $y = \frac { 1 } { x ^ { 2 } }$ and the $x$-axis that lies to the right of the vertical line $x = 3$.\\
(a) Find the area of region $R$.\\
(b) Region $R$ is the base of a solid. For the solid, at each $x$ the cross section perpendicular to the $x$-axis is a rectangle with area given by $x e ^ { x / 5 }$. Find the volume of the solid.\\
(c) Find the volume of the solid generated when the unbounded region $W$ is revolved about the $x$-axis.