3. Let $R$ be the region in the first quadrant enclosed by the hyperbola $x ^ { 2 } - y ^ { 2 } - 9$, the $x$-axis, and the line $x - 5$.
(a) Find the volume of the solid generated by revolving $R$ about the x-axis.
(b) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the solid generated when $R$ is revolved about the line $\mathrm { x } = - 1$.

Solution

(a) Discs:
$$\begin{aligned} V & = \pi \int _ { 3 } ^ { 5 } \left( x ^ { 2 } - 9 \right) d x \\ & = \pi \left[ \frac { 1 } { 3 } x ^ { 3 } - 9 x \right] _ { 3 } ^ { 5 } \\ & = \pi \left[ \left( \frac { 125 } { 3 } - 45 \right) - ( 9 - 27 ) \right] = \frac { 44 } { 3 } \pi \end{aligned}$$
or Shells:
$$\begin{aligned} V & = 2 \pi \int _ { 0 } ^ { 4 } \left( 5 - \sqrt { 9 + y ^ { 2 } } \right) y d y \\ & = 2 \pi \left[ \frac { 5 } { 2 } y ^ { 2 } - \frac { 1 } { 3 } \left( 9 + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } \right] _ { 0 } ^ { 4 } \\ & = 2 \pi \left( 40 - \frac { 125 } { 3 } + \frac { 27 } { 3 } \right) = \frac { 44 } { 3 } \pi \end{aligned}$$
(b) Shells:
$$\begin{aligned} V & = 2 \pi \int _ { 3 } ^ { 5 } ( x + 1 ) y d x \\ & = 2 \pi \int _ { 3 } ^ { 5 } ( x + 1 ) \sqrt { x ^ { 2 } - 9 } d x \end{aligned}$$
or Washers:
$$\begin{aligned} V & = \pi \int _ { 0 } ^ { 4 } \left[ 36 - ( x + 1 ) ^ { 2 } \right] d y \\ & = \pi \int _ { 0 } ^ { 4 } \left[ 36 - \left( \sqrt { 9 + y ^ { 2 } } + 1 \right) ^ { 2 } \right] d y \end{aligned}$$

Distribution of Points

(a)
$$5 : \begin{cases} 2 : & \text { for a correct integrand } \\ 1 : & \text { for appropriate limits } \\ & \text { and } k \pi \\ 1 : & \text { for correct antiderivative } \\ 1 : & \text { for substitution and/or } \\ & \text { evaluation } \end{cases}$$
(b)
$$4 : \left\{ \begin{array} { l } 3 : \text { for a correct integrand } \\ 1 : \text { for appropriate limits } \\ \text { and } k \pi \end{array} \right.$$

1. Let $f$ be the function defined by $f ( x ) = 2 x e ^ { - x }$ for all real numbers $x$.
   (a) Write an equation of the horizontal asymptote for the graph of $f$.
   (b) Find the $x$-coordinate of each critical point of $f$. For each such $x$, determine whether $f ( x )$ is a relative maximum, a relative minimum, or neither.
   (c) For what values of $x$ is the graph of $f$ concave down?
   (d) Using the results found in parts (a), (b), and (c), sketch the graph of $y = f ( x )$ in the $x y$-plane provided below.

Solution

(a) $y = 0$
(b) $f ^ { \prime } ( x ) = 2 \left( - x e ^ { - x } + e ^ { - x } \right)$
$$= 2 e ^ { - x } ( 1 - x )$$
critical point at $x = 1$ relative maximum at $x = 1$
(c) $f ^ { \prime \prime } ( x ) = 2 e ^ { - x } ( - 1 ) + \left( - 2 e ^ { - x } \right) ( 1 - x )$
$$= 2 e ^ { - x } ( x - 2 )$$
Concave down when
$$\begin{aligned} 2 e ^ { - x } ( x - 2 ) & < 0 \\ ( x - 2 ) & < 0 \\ x & < 2 \end{aligned}$$
[Figure]

Distribution of Points

(a) 1: for correct equation

(b)
$$3 : \left\{ \begin{array} { l } 1 : \text { for correct derivative } \\ 1 : \text { for critical value for } f ^ { \prime } \\ 1 : \text { for identifying critical } \\ \text { point as relative maximum } \end{array} \right.$$
(c)
$$2 : \begin{cases} 1 : & \text { for correct } f ^ { \prime \prime } ( x ) \text { for } f ^ { \prime } ( x ) \\ & \text { found in } ( b ) \\ 1 : & \text { for correct interval } \end{cases}$$
(d) 3: for graph consistent with information found in (a), (b), and (c)

Let $R$ be the region bounded by the $x$-axis, the graph of $y = \sqrt{x}$, and the line $x = 4$.
(a) Find the area of the region $R$.
(b) Find the value of $h$ such that the vertical line $x = h$ divides the region $R$ into two regions of equal area.
(c) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(d) The vertical line $x = k$ divides the region $R$ into two regions such that when these two regions are revolved about the $x$-axis, they generate solids with equal volumes. Find the value of $k$.

The figure above shows the graph of the equation $x ^ { \frac { 1 } { 2 } } + y ^ { \frac { 1 } { 2 } } = 2$. Let R be the shaded region between the graph of $x ^ { \frac { 1 } { 2 } } + y ^ { \frac { 1 } { 2 } } = 2$ and the $X$-axis from $x = 0$ to $x = 1$. (a) Find the area of R by setting up and integrating a definite integral. (b) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the solid formed by revolving the region R about the $X$-axis. (c) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the solid formed by revolving the region R about the line $x = 1$.

The shaded region, $R$, is bounded by the graph of $y = x ^ { 2 }$ and the line $y = 4$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the volume of the solid generated by revolving $R$ about the $x$-axis.
(c) There exists a number $k , k > 4$, such that when $R$ is revolved about the line $y = k$, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.

Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $y = e ^ { - x ^ { 2 } } , y = 1 - \cos x$, and the $y$-axis, as shown in the figure above.
(a) Find the area of the region $R$.
(b) Find the volume of the solid generated when the region $R$ is revolved about the $x$-axis.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a square. Find the volume of this solid.

Let $R$ and $S$ be the regions in the first quadrant shown in the figure above. The region $R$ is bounded by the $x$-axis and the graphs of $y = 2 - x^{3}$ and $y = \tan x$. The region $S$ is bounded by the $y$-axis and the graphs of $y = 2 - x^{3}$ and $y = \tan x$.
(a) Find the area of $R$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $S$ is revolved about the $x$-axis.

Let $R$ be the region bounded by the $y$-axis and the graphs of $y = \frac{x^3}{1+x^2}$ and $y = 4 - 2x$, 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, each cross section perpendicular to the $x$-axis is a square. Find the volume of this solid.

Let $R$ be the shaded region bounded by the graphs of $y = \sqrt{x}$ and $y = e^{-3x}$ and the vertical line $x = 1$, 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 horizontal line $y = 1$.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a rectangle whose height is 5 times the length of its base in region $R$. Find the volume of this solid.

Let $R$ be the region enclosed by the graph of $y = \sqrt{x-1}$, the vertical line $x = 10$, and the $x$-axis.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the horizontal line $y = 3$.
(c) Find the volume of the solid generated when $R$ is revolved about the vertical line $x = 10$.

Let $f$ and $g$ be the functions given by $f(x) = 1 + \sin(2x)$ and $g(x) = e^{x/2}$. Let $R$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$.
(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.

Let $R$ be the shaded region bounded by the graph of $y = \ln x$ and the line $y = x - 2$, as shown above.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = -3$.
(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when $R$ is rotated about the $y$-axis.

Let $R$ be the region in the first and second quadrants bounded above by the graph of $y = \frac{20}{1 + x^{2}}$ and below by the horizontal line $y = 2$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated 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. Find the volume of this solid.

Let $R$ be the region bounded by the graphs of $y = \sin ( \pi x )$ and $y = x ^ { 3 } - 4 x$, as shown in the figure above.
(a) Find the area of $R$.
(b) The horizontal line $y = - 2$ splits the region $R$ into two parts. Write, but do not evaluate, an integral expression for the area of the part of $R$ that is below this horizontal line.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a square. Find the volume of this solid.
(d) The region $R$ models the surface of a small pond. At all points in $R$ at a distance $x$ from the $y$-axis, the depth of the water is given by $h ( x ) = 3 - x$. Find the volume of water in the pond.

Let $R$ be the region in the first quadrant enclosed by the graphs of $y = 2x$ and $y = x^{2}$, as shown in the figure above.
(a) Find the area of $R$.
(b) The region $R$ is the base of a solid. For this solid, at each $x$ the cross section perpendicular to the $x$-axis has area $A(x) = \sin\left(\frac{\pi}{2} x\right)$. Find the volume of the solid.
(c) Another solid has the same base $R$. For this solid, the cross sections perpendicular to the $y$-axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.

Let $R$ be the region in the first quadrant bounded by the graph of $y = 2\sqrt{x}$, the horizontal line $y = 6$, and the $y$-axis, as shown in the figure above.
(a) Find the area of $R$.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 7$.
(c) Region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 6$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose height is 3 times the length of its base in region $R$. Write, but do not evaluate, an integral expression that gives the volume of the solid.

Let $R$ be the region enclosed by the graph of $f ( x ) = x ^ { 4 } - 2.3 x ^ { 3 } + 4$ and the horizontal line $y = 4$, as shown in the figure above.
(a) Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = - 2$.
(b) Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is an isosceles right triangle with a leg in $R$. Find the volume of the solid.
(c) The vertical line $x = k$ divides $R$ into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value $k$.

Let $R$ be the region in the first quadrant bounded by the graph of $y = 8 - x^{\frac{3}{2}}$, the $x$-axis, and the $y$-axis.
(a) Find the area of the region $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) The vertical line $x = k$ divides the region $R$ into two regions such that when these two regions are revolved about the $x$-axis, they generate solids with equal volumes. Find the value of $k$.

The shaded region, $R$, is bounded by the graph of $y = x^2$ and the line $y = 4$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated by revolving $R$ about the $x$-axis.
(c) There exists a number $k$, $k > 4$, such that when $R$ is revolved about the line $y = k$, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral expression that can be used to find the value of $k$.

Let $f ( x ) = e ^ { 2 x }$. Let $R$ be the region in the first quadrant bounded by the graph of $f$, the coordinate axes, and the vertical line $x = k$, where $k > 0$. The region $R$ is shown in the figure above. (a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of $R$ in terms of $k$. (b) The region $R$ is rotated about the $x$-axis to form a solid. Find the volume, $V$, of the solid in terms of $k$. (c) The volume $V$, found in part (b), changes as $k$ changes. If $\frac { d k } { d t } = \frac { 1 } { 3 }$, determine $\frac { d V } { d t }$ when $k = \frac { 1 } { 2 }$.

Let $R$ be the region in the first and second quadrants bounded above by the graph of $y = \frac{20}{1 + x^2}$ and below by the horizontal line $y = 2$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated 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. Find the volume of this solid.

Let $R$ be the region in the first quadrant bounded by the graph of $y = 2\sqrt{x}$, the horizontal line $y = 6$, and the $y$-axis.
(a) Find the area of $R$.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 7$.
(c) Region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 6$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose height is 3 times the length of its base in region $R$. Write, but do not evaluate, an integral expression that gives the volume of the solid.

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.