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 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 $R$ 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 in the first quadrant bounded below by the graph of $y = x ^ { 2 }$ and above by the graph of $y = \sqrt { x }$. $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?
(A) 0.129
(B) 0.300
(C) 0.333
(D) 0.700
(E) 1.271

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 $k$.

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height $h$ feet is given by the function $A$, where $A(h)$ is measured in square feet. The function $A$ is continuous and decreases as $h$ increases. Selected values for $A(h)$ are given in the table below.

\begin{tabular}{ c } $h$

(feet)

& 0 & 2 & 5 & 10 \hline

$A ( h )$

(square feet)

& 50.3 & 14.4 & 6.5 & 2.9 \hline \end{tabular}
(a) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.
(b) Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain your reasoning.
(c) The area, in square feet, of the horizontal cross section at height $h$ feet is modeled by the function $f$ given by $f(h) = \frac{50.3}{e^{0.2h} + h}$. Based on this model, find the volume of the tank. Indicate units of measure.
(d) Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rate of 0.26 foot per minute. Using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. Indicate units of measure.

A company designs spinning toys using the family of functions $y = cx\sqrt{4 - x^{2}}$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.
(a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$ for $c = 6$.
(b) It is known that, for $y = cx\sqrt{4 - x^{2}}$, $\frac{dy}{dx} = \frac{c(4 - 2x^{2})}{\sqrt{4 - x^{2}}}$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy?
(c) For another spinning toy, the volume is $2\pi$ cubic inches. What is the value of $c$ for this spinning toy?

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 $g$ 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 $R$ 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.

Let $f(x) = e^{2x}$. 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 $\frac{dk}{dt} = \frac{1}{3}$, determine $\frac{dV}{dt}$ when $k = \frac{1}{2}$.

Let $R$ be the region in the first quadrant bounded above by the graph of $y = \ln ( 3 - x )$, for $0 \leq x \leq 2$. $R$ is the base of a solid for which each cross section perpendicular to the $x$-axis is a square. What is the volume of the solid?
(A) 0.442
(B) 1.029
(C) 1.296
(D) 3.233
(E) 4.071

A company designs spinning toys using the family of functions $y = c x \sqrt { 4 - x ^ { 2 } }$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = c x \sqrt { 4 - x ^ { 2 } }$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.
(a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = c x \sqrt { 4 - x ^ { 2 } }$ for $c = 6$.
(b) It is known that, for $y = c x \sqrt { 4 - x ^ { 2 } } , \frac { d y } { d x } = \frac { c \left( 4 - 2 x ^ { 2 } \right) } { \sqrt { 4 - x ^ { 2 } } }$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy?
(c) For another spinning toy, the volume is $2 \pi$ cubic inches. What is the value of $c$ for this spinning toy?

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.