Content check — Calculus Vol. 2, Chapter 2: Applications of Integration

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 2: Applications of Integration

272 problems · 70 figures

Checkpoint 2.1 §2.1 source ↗

R

is the region bounded by the graphs of the functions

f (x) = \frac{x}{2} + 5

and

g (x) = x + \frac{1}{2}

over the interval

[1, 5],

find the area of region

R .

Answer

12

units²

Checkpoint 2.2 §2.1 source ↗

If R is the region bounded above by the graph of the function

f (x) = x

and below by the graph of the function

g (x) = x^{4},

find the area of region

R .

Answer

\frac{3}{10}

unit²

Checkpoint 2.3 §2.1 source ↗

If R is the region between the graphs of the functions

f (x) = sin x

and

g (x) = cos x

over the interval

[π / 2, 2 π],

find the area of region

R .

Answer

2 + 2 \sqrt{2}

units²

Checkpoint 2.4 §2.1 source ↗

Consider the region depicted in the following figure. Find the area of $R .$

This figure is has two graphs in the first quadrant. They are the functions f(x) = squareroot of x and g(x)= 3/2 – x/2. In between these graphs is a shaded region, bounded to the left by f(x) and to the right by g(x). All of which is above the x-axis. The shaded area is between x=0 and x=3.

Answer

\frac{5}{3}

units²

Checkpoint 2.5 §2.1 source ↗

Let’s revisit the checkpoint associated with Example 2.4, only this time, let’s integrate with respect to $y .$ Let $R$ be the region depicted in the following figure. Find the area of $R$ by integrating with respect to $y .$

Answer

\frac{5}{3}

units²

2-1 §2.1 source ↗

For the following exercises, determine the area of the region between the two curves in the given figure by integrating over the $x -axis .$

$y =$ $x^{2} -$ $3 and y =$ $1$

This figure is has two graphs. They are the functions f(x) = x^2-3and g(x)=1. In between these graphs is a shaded region, bounded above by g(x) and below by f(x). The shaded area is between x=-2 and x=2.

Answer

\frac{32}{3}

2-3 §2.1 source ↗

For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the $x -axis .$ Note that you will have two integrals to solve.

$y = x^{3}$ and $y = x^{2} + x$

This figure is has two graphs. They are the functions f(x) = x^3 and g(x)= x^2+x. These graphs intersect twice. The regions between the intersections are shaded. The first region is bounded above by f(x) and below by g(x). The second region is bounded above by g(x) and below by f(x).

Answer

\frac{13}{12}

2-5 §2.1 source ↗

For the following exercises, determine the area of the region between the two curves by integrating over the $y -axis .$

$x = y^{2} and x = 9$

Answer

36

2-7 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x -axis .$

$y =$ $x^{2} and y =$ $− x^{2} +$ $18 x$

Answer

This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=-x^2+18x. The region between the graphs is shaded, bounded above by g(x) and below by f(x). It is in the first quadrant.

243 square units

2-9 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x -axis .$

$y = cos x$ and $y = {cos}^{2} x$ on $x = [− π, π]$

Answer

This figure is has two graphs. They are the functions y=cos(x) and y=cos^2(x). The graphs are periodic and resemble waves. There are four regions created by intersections of the curves. The areas are shaded.

2-11 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x -axis .$

$y =$ $e^{x},$ $y =$ $e^{− x},$ $x =$ $−1 and x =$ $1$

Answer

This figure is has two graphs. They are the functions f(x)=e^x and g(x)=e^-x. There are two shaded regions. In the second quadrant the region is bounded by x=-1, g(x) above and f(x) below. The second region is in the first quadrant and is bounded by f(x) above, g(x) below, and x=1.

$\frac{2 {(e - 1)}^{2}}{e}$

2-13 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x -axis .$

$y =$ $| x | and y =$ $x^{2}$

Answer

This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=absolute value of x. There are two shaded regions. The first region is in the second quadrant and is between g(x) above and f(x) below. The second region is in the first quadrant and is bounded above by g(x) and below by f(x).

$\frac{1}{3}$

2-15 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.

$y =$ $12 -$ $x,$ $y =$ $\sqrt{x},$ $and y =$ $1$

Answer

This figure is has three graphs. They are the functions f(x)=squareroot of x, y=12-x, and y=1. The region between the graphs is shaded, bounded above and to the left by f(x), above and to the right by the line y=12-x, and below by the line y=1. It is in the first quadrant.

$\frac{34}{3}$

2-17 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.

$y =$ $x^{3} and y =$ $x^{2} -$ $2 x$ over $x = [−1, 1]$

Answer

This figure is has two graphs. They are the functions f(x)=x^3 and g(x)=x^2-2x. There are two shaded regions between the graphs. The first region is bounded to the left by the line x=-2, above by g(x) and below by f(x). The second region is bounded above by f(x), below by g(x) and to the right by the line x=2.

$\frac{5}{2}$

2-19 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.

$y = x^{3} + 3 x$ and $y = 4 x$

Answer

This figure is has two graphs. They are the functions f(x)=x^3+3x and g(x)=4x. There are two shaded regions between the graphs. The first region is bounded above by f(x) and below by g(x). The second region is bounded above by g(x), below by f(x).

$\frac{1}{2}$

2-21 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y -axis .$

$x =$ $2 y and x =$ $y^{3} -$ $y$

Answer

$\frac{9}{2}$

2-23 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y -axis .$

$y^{2} =$ $x and x =$ $y +$ $2$

Answer

This figure is has two graphs. They are the equations x=y+2 and y^2=x. The graphs intersect, forming a region in between them

$\frac{9}{2}$

2-25 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y -axis .$

$x =$ $sin y,$ $x =$ $cos (2 y),$ $y =$ $π / 2,$ $and y =$ $− π / 2$

Answer

This figure is has two graphs. They are the equations x=cos(y) and x=sin(y). The graphs intersect, forming two regions bounded above by the line y=pi/2 and below by the line y=-pi/2.

$\frac{3 \sqrt{3}}{2}$

2-27 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

$y =$ $x e^{x},$ $y =$ $e^{x},$ $x =$ $0,$ $and x =$ $1$

Answer

This figure is has two graphs. They are the equations y=xe^x and y=e^x. The graphs intersect, forming a region in between them in the first quadrant.

$e −2$

2-29 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

$x =$ $y^{3} +$ $2 y^{2} +$ $1 and x =$ $− y^{2} +$ $1$

Answer

This figure is has two graphs. They are the equations x=-y^2+1 and x=y^3+2y^2. The graphs intersect, forming two regions in between them.

$\frac{27}{4}$

2-31 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

$y =$ $4 -$ $3 x and y =$ $\frac{1}{x}$

Answer

This figure is has two graphs. They are the equations y=4-3x and y=1/x. The graphs intersect, having region between them shaded. The region is in the first quadrant.

$\frac{4}{3} - ln (3)$

2-33 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

$y =$ $x^{2} -$ $3 x +$ $2 and y =$ $x^{3} -$ $2 x^{2} -$ $x +$ $2$

Answer

$\frac{1}{2}$

2-35 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

$y +$ $y^{3} =$ $x and 2 y =$ $x$

Answer

This figure is has two graphs. They are the equations 2y=x and y+y^3=x. The graphs intersect, forming two regions. The regions are shaded.

$\frac{1}{2}$

2-37 §2.1 source ↗

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

$y =$ ${cos}^{−1} x,$ $y =$ ${sin}^{−1} x,$ $x =$ $−1,$ $and x =$ $1$

Answer

This figure is has two graphs. They are the equations y=arccos(x) and y=arcsin (x). The graphs intersect, forming two regions. The first region is bounded to the left by x=-1. The second region is bounded to the right by x=1. Both regions are shaded.

$−2 (\sqrt{2} - π)$

2-39 §2.1 source ↗

For the following exercises, find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.

[T] $y =$ $x^{2} and y =$ $\sqrt{1 - x^{2}}$

Answer

1.067

2-41 §2.1 source ↗

[T] $x =$ $\sqrt{4 - y^{2}} and y^{2} =$ $1 +$ $x^{2}$

Answer

1.605

2-43 §2.1 source ↗

[T] $y =$ ${sin}^{3} x +$ $2,$ $y =$ $tan x,$ $x =$ $−1.5,$ $and x =$ $1.5$

Answer

7.523

2-45 §2.1 source ↗

[T] $y =$ $\sqrt{1 - x^{2}} and y =$ $x^{2} +$ $2 x +$ $1$

Answer

\frac{3 π - 4}{12}

2-47 §2.1 source ↗

[T] $y =$ $cos x,$ $y =$ $e^{x},$ $x =$ $− π,$ $and x =$ $0$

Answer

1.429

2-49 §2.1 source ↗

A factory selling cell phones has a marginal cost function $C (x) =$ $0.01 x^{2} -$ $3 x +$ $229,$ where $x$ represents the number of cell phones, $C$ is the marginal cost, in dollars, of selling $x$ phones, and a marginal revenue function given by $R (x) = 429 - 2 x$ , where $R$ is the revenue, in dollars, earned by selling $x$ cell phonesFind the area between the graphs of these curves and $x = 0 .$ What does this area represent?

Answer

$ 33,333.33

total profit for

200

cell phones sold

2-51 §2.1 source ↗

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H (t) =$ $1 -$ $cos ((π t) / 2)$ whereas the speed of the tortoise is $T (t) =$ $0.1032 t$ where $t$ is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time $t = 0$ to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

Answer

3.32

mi represents how far ahead the hare is from the tortoise

2-53 §2.1 source ↗

For the following exercises, find the area between the curves by integrating with respect to $x$ and then with respect to $y .$ Is one method easier than the other? Do you obtain the same answer?

$y =$ $x^{2} +$ $2 x +$ $1 and y =$ $− x^{2} -$ $3 x +$ $4$

Answer

\frac{343}{24}

2-55 §2.1 source ↗

For the following exercises, find the area between the curves by integrating with respect to $x$ and then with respect to $y .$ Is one method easier than the other? Do you obtain the same answer?

$x =$ $y^{2} -$ $2 and x =$ $2 y$

Answer

4 \sqrt{3}

2-57 §2.1 source ↗

For the following exercises, solve using calculus, then check your answer with geometry.

Find the area between the perimeter of the unit circle and the triangle created from $y =$ $2 x +$ $1,$ $y =$ $1 -$ $2 x$ and $y = - \frac{3}{5},$ as seen in the following figure. Is there a way to solve this without using calculus?

This figure is the graph of a circle centered at the origin with radius of 1. There are three lines intersecting the circle. The lines intersect the circle at three points to form a triangle within the circle.

Answer

π - \frac{32}{25}

Checkpoint 2.7 §2.2 source ↗

Use the method of slicing to find the volume of the solid of revolution formed by revolving the region between the graph of the function $f (x) = 1 / x$ and the $x -axis$ over the interval $[1, 2]$ around the $x -axis .$ See the following figure.

This figure has two graphs. The first graph is the curve f(x)=1/x. It is a decreasing curve, above the x-axis in the first quadrant. The graph has a shaded region under the curve between x=1 and x=2. The second graph is the curve f(x)=1/x in the first quadrant. Also, underneath this graph, there is a solid between x=1 and x=2 that has been formed by rotating the region from the first graph around the x-axis.

Answer

\frac{π}{2}

Checkpoint 2.8 §2.2 source ↗

Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of

f (x) = \sqrt{4 - x}

and the

x -axis

over the interval

[0, 4]

around the

x -axis .

Answer

8 π

units³

Checkpoint 2.9 §2.2 source ↗

Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of

g (y) = y

and the

y -axis

over the interval

[1, 4]

around the

y -axis .

Answer

21 π

units³

Checkpoint 2.10 §2.2 source ↗

Find the volume of a solid of revolution formed by revolving the region bounded by the graphs of

f (x) = \sqrt{x}

and

g (x) = 1 / x

over the interval

[1, 3]

around the

x -axis .

Answer

\frac{10 π}{3}

units³

Checkpoint 2.11 §2.2 source ↗

Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of

f (x) = x + 2

and below by the

x -axis

over the interval

[0, 3]

around the line

y = −1 .

Answer

60 π

units³

2-63 §2.2 source ↗

For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A pyramid with height 6 units and square base of side 2 units, as pictured here.

This figure is a pyramid with base width of 2 and height of 6 units.

Answer

8 units³

2-65 §2.2 source ↗

For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A tetrahedron with a base side of 4 units, as seen here.

This figure is an equilateral triangle with side length of 4 units.

Answer

\frac{32}{3 \sqrt{2}}

units³

2-67 §2.2 source ↗

For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A cone of radius $r$ and height $h$ has a smaller cone of radius $r / 2$ and height $h / 2$ removed from the top, as seen here. The resulting solid is called a frustum.

This figure is a 3-dimensional graph of an upside down cone. The cone is inside of a rectangular prism that represents the xyz coordinate system. the radius of the bottom of the cone is “r” and the radius of the top of the cone is labeled “r/2”.

Answer

\frac{7}{24} π r^{2} h

units³

2-69 §2.2 source ↗

For the following exercises, draw an outline of the solid and find the volume using the slicing method.

The base is a triangle with vertices $(0, 0),$ $(1, 0),$ and $(0, 1) .$ Slices perpendicular to the x-axis are semicircles.

Answer

This figure shows the x-axis and the y-axis with a line starting on the x-axis at (1,0) and ending on the y-axis at (0,1). Perpendicular to the xy-plane are 4 shaded semi-circles with their diameters beginning on the x-axis and ending on the line, decreasing in size away from the origin.

$\frac{π}{24}$ units³

2-71 §2.2 source ↗

For the following exercises, draw an outline of the solid and find the volume using the slicing method.

The base is the region under the parabola $y = 1 - x^{2}$ and above the $x -axis .$ Slices perpendicular to the $y -axis$ are squares.

Answer

This figure shows the x-axis and the y-axis in 3-dimensional perspective. On the graph above the x-axis is a parabola, which has its vertex at y=1 and x-intercepts at (-1,0) and (1,0). There are 3 square shaded regions perpendicular to the x y plane, which touch the parabola on either side, decreasing in size away from the origin.

$2$ units³

2-73 §2.2 source ↗

For the following exercises, draw an outline of the solid and find the volume using the slicing method.

The base is the area between $y = x$ and $y = x^{2} .$ Slices perpendicular to the x-axis are semicircles.

Answer

This figure is a graph with the x and y axes diagonal to show 3-dimensional perspective. On the first quadrant of the graph are the curves y=x, a line, and y=x^2, a parabola. They intersect at the origin and at (1,1). Several semicircular-shaped shaded regions are perpendicular to the x y plane, which go from the parabola to the line and perpendicular to the line.

$\frac{π}{240}$ units³

2-75 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$y =$ $2 x^{2},$ $x =$ $0,$ $x =$ $4,$ $and y =$ $0$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^2, below by the x-axis, and to the right by the vertical line x=4.

$\frac{4096 π}{5}$ units³

2-77 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$y =$ $x^{4},$ $x =$ $0,$ $and y =$ $1 for x \geq 0$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=1, below by the curve y=x^4, and to the left by the y-axis.

$\frac{8 π}{9}$ units³

2-79 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$y =$ $sin x,$ $y =$ $cos x,$ $and x =$ $0$

Answer

This figure is a shaded region bounded above by the curve y=cos(x), below to the left by the y-axis and below to the right by y=sin(x). The shaded region is in the first quadrant.

$\frac{π}{2}$ units³

2-81 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the x-axis.

$x^{2} -$ $y^{2} =$ $9 and x +$ $y =$ $9,$ $y =$ $0 and x =$ $0$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line x + y=9, below by the x-axis, to the left by the y-axis, and to the left by the curve x^2-y^2=9.

$207 π$ units³

2-83 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$y =$ $2 x^{3},$ $x =$ $0,$ $x =$ $1,$ $and y =$ $0$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^3, below by the x-axis, and to the right by the line x=1.

$\frac{4 π}{5}$ units³

2-85 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$y =$ $\sqrt{4 - x^{2}},$ $y =$ $0,$ $and x =$ $0$

Answer

This figure is a graph in the first quadrant. It is a quarter of a circle with center at the origin and radius of 2. It is shaded on the inside.

$\frac{16 π}{3}$ units³

2-87 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$x =$ $sec (y) and y =$ $\frac{π}{4},$ $y =$ $0 and x =$ $0$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=pi/4, to the right by the curve x=sec(y), below by the x-axis, and to the left by the y-axis.

$π$ units³

2-89 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.

$y =$ $4 -$ $x,$ $y =$ $x,$ $and x =$ $0$

Answer

This figure is a graph in the first quadrant. It is a shaded triangle bounded above by the line y=4-x, below by the line y=x, and to the left by the y-axis.

$\frac{16 π}{3}$ units³

2-91 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y =$ $x^{2} and y =$ $x +$ $2$

Answer

This figure is a graph above the x-axis. It is a shaded region bounded above by the line y=x+2, and below by the parabola y=x^2.

$\frac{72 π}{5}$ units³

2-93 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y =$ $4 -$ $x^{2} and y =$ $2 -$ $x$

Answer

This figure is a shaded region bounded above by the curve y=4-x^2 and below by the line y=2-x.

$\frac{108 π}{5}$ units³

2-95 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y = \sqrt{x} and y = x^{2}$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=squareroot(x), below by the curve y=x^2.

$\frac{3 π}{10}$ units³

2-97 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the x-axis.

$y =$ $\sqrt{1 + x^{2}} and y =$ $\sqrt{4 - x^{2}}$

Answer

This figure is a shaded region bounded above by the curve y=squareroot(4-x^2) and, below by the curve y=squareroot(1+x^2).

$2 \sqrt{6} π$ units³

2-99 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.

$y =$ $x +$ $2,$ $y =$ $2 x -$ $1,$ $and x =$ $0$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=x+2, below by the line y=2x-1, and to the left by the y-axis.

$9 π$ units³

2-101 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.

$x =$ $e^{2 y},$ $x =$ $y^{2},$ $y =$ $0,$ $and y =$ $ln (2)$

Answer

This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=ln(2), below by the x-axis, to the left by the curve x=y^2, and to the right by the curve x=e^(2y).

$\frac{π}{20} (75 - 4 {ln}^{5} (2))$ units³

2-103 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.

Yogurt containers can be shaped like frustums. Rotate the line $y = \frac{1}{m} x$ around the y-axis to find the volume between $y = a and y = b .$

This figure has two parts. The first part is a solid cone. The base of the cone is wider than the top. It is shown in a 3-dimensional box. Underneath the cone is an image of a yogurt container with the same shape as the figure.

Answer

\frac{m^{2} π}{3} (b^{3} - a^{3})

units³

2-105 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.

Rotate the ellipse $(x^{2} / a^{2}) +$ $(y^{2} / b^{2}) =$ $1$ around the y-axis to approximate the volume of a football.

Answer

\frac{4 a^{2} b π}{3}

units³

2-107 §2.2 source ↗

For the following exercises, draw the region bounded by the curves. Then, use the washer method to find the volume when the region is revolved around the y-axis.

What is the volume of the Bundt cake that comes from rotating $y = sin x$ around the y-axis from $x = 0$ to $x = π ?$

This figure is a graph of a 3-dimensional solid. It is round, bigger towards the bottom. It has a hole in the center that progressively gets smaller towards the bottom. Next to the graph is an image of a bundt cake, resembling the solid.

Answer

2 π^{2}

units³

2-109 §2.2 source ↗

For the following exercises, find the volume of the solid described.

The base is the region enclosed by the generic ellipse $(x^{2} / a^{2}) +$ $(y^{2} / b^{2}) =$ $1 .$ Slices perpendicular to the x-axis are semicircles.

Answer

\frac{2 a b^{2} π}{3}

units³

2-111 §2.2 source ↗

For the following exercises, find the volume of the solid described.

Find the volume common to two spheres of radius $r$ with centers that are $2 h$ apart, as shown here.

This figure has two circles that intersect. Both circles have radius “r”. There is a line segment from one center to the other. In the middle of the intersection of the circles is point “h”. It is on the line segment.

Answer

V =

\frac{2 π}{3} (2 r + h) {r - h}^{2}

units³

2-113 §2.2 source ↗

For the following exercises, find the volume of the solid described.

Find the volume of a sphere of radius $R$ with a cap of height $h$ removed from the top, as seen here.

This figure is a sphere with a top portion removed. The radius of the sphere is “R”. The distance from the center to where the top portion is removed is “R-h”.

Answer

\frac{π}{3} (h + R) {(h - 2 R)}^{2}

units³

Checkpoint 2.12 §2.3 source ↗

Define R as the region bounded above by the graph of

f (x) = x^{2}

and below by the x-axis over the interval

[1, 2] .

Find the volume of the solid of revolution formed by revolving

R

around the

y -axis .

Answer

\frac{15 π}{2}

units³

Checkpoint 2.13 §2.3 source ↗

Define

R

as the region bounded above by the graph of

f (x) = 3 x - x^{2}

and below by the

x -axis

over the interval

[0, 2] .

Find the volume of the solid of revolution formed by revolving

R

around the

y -axis .

Answer

8 π

units³

Checkpoint 2.14 §2.3 source ↗

Define

Q

as the region bounded on the right by the graph of

g (y) = 3 / y

and on the left by the

y -axis

for

y \in [1, 3] .

Find the volume of the solid of revolution formed by revolving

Q

around the

x -axis .

Answer

12 π

units³

Checkpoint 2.15 §2.3 source ↗

Define

R

as the region bounded above by the graph of

f (x) = x^{2}

and below by the

x -axis

over the interval

[0, 1] .

Find the volume of the solid of revolution formed by revolving

R

around the line

x = −2 .

Answer

\frac{11 π}{6}

units³

Checkpoint 2.16 §2.3 source ↗

Define

R

as the region bounded above by the graph of

f (x) = x

and below by the graph of

g (x) = x^{2}

over the interval

[0, 1] .

Find the volume of the solid of revolution formed by revolving

R

around the

y -axis .

Answer

\frac{π}{6}

units³

Checkpoint 2.17 §2.3 source ↗

Select the best method to find the volume of a solid of revolution generated by revolving the given region around the

x -axis,

and set up the integral to find the volume (do not evaluate the integral): the region bounded by the graphs of

y = 2 - x^{2}

and

y = x^{2} .

Answer

Use the method of washers;

V =

\int_{−1}^{1} π [{(2 - x^{2})}^{2} - {(x^{2})}^{2}] d x

2-115 §2.3 source ↗

For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.

[T] Bounded by the curves $y =$ $3 x,$ $y =$ $0,$ $and x =$ $3$ rotated around the $y -axis .$

Answer

This figure is a graph in the first quadrant. It is the line y=3x. Under the line and above the x-axis there is a shaded region. The region is bounded to the right at x=3.

$54 π$ units³

2-117 §2.3 source ↗

[T] Bounded by the curves $y =$ $3 x,$ $y =$ $0,$ $and x =$ $3$ rotated around the $x -axis .$

Answer

$81 π$ units³

2-119 §2.3 source ↗

[T] Bounded by the curves $y =$ $2 x^{3},$ $y =$ $0,$ $and x =$ $2$ rotated around the $x -axis .$

Answer

This figure is a graph in the first quadrant. It is the increasing curve y=2x^3. Under the curve and above the x-axis there is a shaded region. The region is bounded to the right at x=2.

$\frac{512 π}{7}$ units³

2-121 §2.3 source ↗

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$

$y =$ $5 x^{3},$ $x =$ $0,$ $and x =$ $1$

Answer

2 π

units³

2-123 §2.3 source ↗

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$

$y =$ $\sqrt{1 - x^{2}},$ $x =$ $0,$ $and x =$ $1$

Answer

\frac{2 π}{3}

units³

2-125 §2.3 source ↗

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$

$y =$ $sin x^{2},$ $x =$ $0,$ $and x =$ $\sqrt{π}$

Answer

2 π

units³

2-127 §2.3 source ↗

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$

$y =$ $\sqrt{x},$ $x =$ $0,$ $and x =$ $1$

Answer

\frac{4 π}{5}

units³

2-129 §2.3 source ↗

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x -axis$ and are rotated around the $y -axis .$

$y =$ $5 x^{3} -$ $2 x^{4},$ $x =$ $0,$ $and x =$ $2$

Answer

\frac{64 π}{3}

units³

2-131 §2.3 source ↗

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$

$y =$ $x^{2},$ $x =$ $0,$ $x =$ $2$ and the x-axis

Answer

\frac{32 π}{5}

units³

2-133 §2.3 source ↗

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$

$y =$ $\frac{2}{x^{2}},$ $x =$ $1,$ $x =$ $2,$ and the x-axis

Answer

\frac{7 π}{6}

2-135 §2.3 source ↗

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$

$x =$ $\frac{1 + y^{2}}{y},$ $y =$ $1,$ $y =$ $4,$ and the y-axis

Answer

48 π

2-137 §2.3 source ↗

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$

$x =$ $y^{3} - 2 y^{2},$ $x =$ $0,$ $x =$ $9$

Answer

\frac{308 π}{5}

2-139 §2.3 source ↗

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and $y = 0$ around the $x -axis .$

$x =$ $\sqrt[3]{27 y} and x =$ $\frac{3 y}{4}$

Answer

\frac{512 π}{7}

2-141 §2.3 source ↗

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

$y =$ $x^{3},$ $x =$ $0,$ $and y =$ $8$ rotated around the $y -axis .$

Answer

\frac{8 π}{5}

2-143 §2.3 source ↗

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

$y =$ $\sqrt{x},$ $y =$ $0,$ $and x =$ $1$ rotated around the line $x = 2 .$

Answer

\frac{28 π}{15}

units³

2-145 §2.3 source ↗

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

$y = \sqrt{x} and y = x^{2}$ rotated around the $y -axis .$

Answer

\frac{3 π}{10}

units³

2-147 §2.3 source ↗

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

$x =$ $y^{3},$ $x =$ $\frac{1}{y},$ $x =$ $1,$ $and x =$ $2$ rotated around the $x -axis .$

Answer

π \frac{6 \cdot 2^{2 / 3}}{5} - \frac{11}{10} =

\frac{π}{10} 12 \cdot 2^{2 / 3} - 11 \approx

2.5286

units³

2-149 §2.3 source ↗

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

[T] Left of $x = sin (π y),$ right of $y = x,$ around the $y -axis .$

Answer

0.9876

units³

2-151 §2.3 source ↗

For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.

[T] $y =$ $cos (π x),$ $y =$ $sin (π x),$ $x =$ $\frac{1}{4},$ $and x =$ $\frac{5}{4}$ rotated around the $y -axis .$ This exercise requires advanced technique. You may use technology to perform the integration.

Answer

This figure is a graph. On the graph are two curves, y=cos(pi times x) and y=sin(pi times x). They are periodic curves resembling waves. The curves intersect in the first quadrant and also the fourth quadrant. The region between the two points of intersection is shaded.

$3 \sqrt{2}$ units³

2-153 §2.3 source ↗

[T] $y =$ $x^{2} -$ $2 x,$ $x =$ $2,$ $and x =$ $4$ rotated around the $x -axis .$

Answer

This figure is a graph in the first quadrant. It is the parabola y=x^2-2x. . Under the curve and above the x-axis there is a shaded region. The region begins at x=2 and is bounded to the right at x=4.

$\frac{496 π}{15}$ units³

2-155 §2.3 source ↗

[T] $y =$ $3 x^{3} -$ $2,$ $y =$ $x,$ $and x =$ $2$ rotated around the $y -axis .$

Answer

This figure is a graph in the first quadrant. There are two curves on the graph. The first curve is y=3x^2-2 and the second curve is y=x. Between the curves there is a shaded region. The region begins at x=1 and is bounded to the right at x=2.

$\frac{398 π}{15}$ units³

2-157 §2.3 source ↗

[T] $x =$ $y^{2},$ $x =$ $y^{2} -$ $2 y +$ $1,$ $and x =$ $2$ rotated around the $y -axis .$

Answer

This figure is a graph. There are two curves on the graph. The first curve is x=y^2-2y+1 and is a parabola opening to the right. The second curve is x=y^2 and is a parabola opening to the right. Between the curves there is a shaded region. The shaded region is bounded to the right at x=2.

$15.9074$ units³

2-159 §2.3 source ↗

For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.

Use the method of shells to find the volume of a cone with radius $r$ and height $h .$

This figure has two images. The first is an upside-down cone with radius r and height h. The second is an ice cream cone.

Answer

\frac{1}{3} π r^{2} h

units³

2-161 §2.3 source ↗

For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.

Use the method of shells to find the volume of a cylinder with radius $r$ and height $h .$

This figure has two images. The first is a cylinder with radius r and height h. The second is a cylindrical candle.

Answer

π r^{2} h

units³

2-163 §2.3 source ↗

For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.

Consider the region enclosed by the graphs of $y =$ $f (x),$ $y =$ $1 +$ $f (x),$ $x =$ $0,$ $y =$ $0,$ and $x = a > 0 .$ What is the volume of the solid generated when this region is rotated around the $y -axis ?$ Assume that the function is defined over the interval $[0, a] .$

Answer

π a^{2}

units³

Checkpoint 2.18 §2.4 source ↗

Let

f (x) =

(4 / 3) x^{3 / 2} .

Calculate the arc length of the graph of

f (x)

over the interval

[0, 1] .

Round the answer to three decimal places.

Answer

\frac{1}{6} (5 \sqrt{5} - 1) \approx

1.697

Checkpoint 2.19 §2.4 source ↗

Let

f (x) = sin x .

Calculate the arc length of the graph of

f (x)

over the interval

[0, π] .

Use a computer or calculator to approximate the value of the integral.

Answer

Arc Length \approx

3.8202

Checkpoint 2.20 §2.4 source ↗

Let

g (y) = 1 / y .

Calculate the arc length of the graph of

g (y)

over the interval

[1, 4] .

Use a computer or calculator to approximate the value of the integral.

Answer

Arc Length =

3.15018

Checkpoint 2.21 §2.4 source ↗

Let

f (x) = \sqrt{1 - x}

over the interval

[0, 1 / 2] .

Find the surface area of the surface generated by revolving the graph of

f (x)

around the

x -axis .

Round the answer to three decimal places.

Answer

\frac{π}{6} (5 \sqrt{5} - 3 \sqrt{3}) \approx

3.133

Checkpoint 2.22 §2.4 source ↗

Let

g (y) = \sqrt{9 - y^{2}}

over the interval

y \in [0, 2] .

Find the surface area of the surface generated by revolving the graph of

g (y)

around the

y -axis .

Answer

12 π

2-165 §2.4 source ↗

For the following exercises, find the length of the functions over the given interval.

$y =$ $5 x from x =$ $0 to x =$ $2$

Answer

2 \sqrt{26}

2-167 §2.4 source ↗

For the following exercises, find the length of the functions over the given interval.

$x =$ $4 y from y =$ $−1 to y =$ $1$

Answer

2 \sqrt{17}

2-169 §2.4 source ↗

For the following exercises, find the length of the functions over the given interval.

Find the surface area of the volume generated when the curve $y = \sqrt{x}$ revolves around the $x -axis$ from $(1, 1)$ to $(4, 2),$ as seen here.

This figure is a surface. It has been formed by rotating the curve y=squareroot(x) about the x-axis. The surface is inside of a cube to show 3-dimensions.

Answer

\frac{π}{6} (17 \sqrt{17} - 5 \sqrt{5})

2-171 §2.4 source ↗

For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$y = x^{3 / 2}$ from $(0, 0) to (1, 1)$

Answer

\frac{13 \sqrt{13} - 8}{27}

2-173 §2.4 source ↗

For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$y =$ $\frac{1}{3} {(x^{2} + 2)}^{3 / 2}$ from $x = 0 to x = 1$

Answer

\frac{4}{3}

2-175 §2.4 source ↗

For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

[T] $y = e^{x}$ on $x = 0$ to $x = 1$

Answer

2.0035

2-177 §2.4 source ↗

For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$y = \frac{x^{4}}{4} + \frac{1}{8 x^{2}}$ from $x = 1 to x = 2$

Answer

\frac{123}{32}

2-179 §2.4 source ↗

For the following exercises, find the lengths of the functions of $x$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$y =$ $\frac{1}{27} {(9 x^{2} + 6)}^{3 / 2}$ from $x = 0 to x = 2$

Answer

10

2-181 §2.4 source ↗

For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$y = \frac{5 - 3 x}{4}$ from $y = 0$ to $y = 4$

Answer

\frac{20}{3}

2-183 §2.4 source ↗

For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$x = 5 y^{3 / 2}$ from $y = 0$ to $y = 1$

Answer

\frac{1}{675} (229 \sqrt{229} - 8)

2-185 §2.4 source ↗

For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

$x = \sqrt{y}$ from $y = 0 to y = 1$

Answer

\frac{1}{8} (4 \sqrt{5} + ln (9 + 4 \sqrt{5}))

2-187 §2.4 source ↗

For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

[T] $x = tan y$ from $y = 0$ to $y = \frac{3}{4}$

Answer

1.201

2-189 §2.4 source ↗

For the following exercises, find the lengths of the functions of $y$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.

[T] $x = 4^{y}$ from $y = 0 to y = 2$

Answer

15.2341

2-191 §2.4 source ↗

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $x -axis .$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y = \sqrt{x}$ from $x = 2$ to $x = 6$

Answer

\frac{49 π}{3}

2-193 §2.4 source ↗

$y = 7 x$ from $x = −1 to x = 1$

Answer

70 π \sqrt{2}

2-195 §2.4 source ↗

$y = \sqrt{4 - x^{2}}$ from $x = 0 to x = 2$

Answer

8 π

2-197 §2.4 source ↗

$y = 5 x$ from $x = 1 to x = 5$

Answer

120 π \sqrt{26}

2-199 §2.4 source ↗

For the following exercises, find the surface area of the volume generated when the following curves revolve around the $y -axis .$ If you cannot evaluate the integral exactly, use your calculator to approximate it.

$y = x^{2}$ from $x = 0 to x = 2$

Answer

\frac{π}{6} (17 \sqrt{17} - 1)

2-201 §2.4 source ↗

$y = x + 1$ from $x = 0 to x = 3$

Answer

9 \sqrt{2} π

2-203 §2.4 source ↗

$y = \sqrt[3]{x}$ from $x = 1 to x = 27$

Answer

\frac{10 \sqrt{10} π}{27} (73 \sqrt{73} - 1)

2-205 §2.4 source ↗

[T] $y = \frac{1}{\sqrt{x}}$ from $x = 1$ to $x = 3$

Answer

25.645

2-207 §2.4 source ↗

The base of a lamp is constructed by revolving a quarter circle $y = \sqrt{2 x - x^{2}}$ around the $y -axis$ from $x = 1$ to $x = 2,$ as seen here. Create an integral for the surface area of this curve and compute it.

This figure is a surface. It is half of a torus created by rotating the curve y=squareroot(2x-x^2) about the x-axis.

Answer

π (2 + π)

2-209 §2.4 source ↗

[T] A lampshade is constructed by rotating $y = 1 / x$ around the $x -axis$ from $y = 1$ to $y = 2,$ as seen here. Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places.

This figure has two images. The first is similar to a frustum of a cone with edges bending inwards. The second is a lamp shade.

Answer

10.5017

2-211 §2.4 source ↗

[T] You are building a bridge that will span $10$ ft. You intend to add decorative rope in the shape of $y =$ $5 | sin ((x π) / 5) |,$ where $x$ is the distance in feet from one end of the bridge. Find out how much rope you need to buy, measured in a whole number of feet.

Answer

24

2-213 §2.4 source ↗

For the following exercises, find the exact arc length for the following problems over the given interval.

[T] Draw graphs of $y = x^{2},$ $y = x^{6},$ and $y = x^{10} .$ For $y = x^{n},$ as $n$ increases, formulate a prediction on the arc length from $(0, 0)$ to $(1, 1) .$ Now, compute the lengths of these three functions and determine whether your prediction is correct.

Answer

2

2-215 §2.4 source ↗

For the following exercises, find the exact arc length for the following problems over the given interval.

Solve for the length of $x = y^{2}$ from $(0, 0) to (1, 1) .$ Show that $x = (1 / 2) y^{2}$ from $(0, 0)$ to $(2, 2)$ is twice as long. Graph both functions and explain why this is so.

Answer

Answers may vary

2-217 §2.4 source ↗

For the following exercises, find the exact arc length for the following problems over the given interval.

Explain why the surface area is infinite when $y = 1 / x$ is rotated around the $x -axis$ for $1 \leq x < \infty,$ but the volume is finite.

Answer

For more information, look up Gabriel’s Horn.

Checkpoint 2.23 §2.5 source ↗

Consider a thin rod oriented on the x-axis over the interval

[1, 3] .

If the density of the rod is given by

ρ (x) = 2 x^{2} + 3,

what is the mass of the rod?

Answer

70 / 3

Checkpoint 2.24 §2.5 source ↗

Let

ρ (x) = 3 x + 2

represent the radial density of a disk. Calculate the mass of a disk of radius 2.

Answer

24 π

Checkpoint 2.25 §2.5 source ↗

Suppose it takes a force of

8

lb to stretch a spring

6

in. from the equilibrium position. How much work is done to stretch the spring

1

ft from the equilibrium position?

Answer

8

ft-lb

Checkpoint 2.26 §2.5 source ↗

A tank is in the shape of an inverted cone, with height

10

ft and base radius 6 ft. The tank is filled to a depth of 8 ft to start with, and water is pumped over the upper edge of the tank until 3 ft of water remain in the tank. How much work is required to pump out that amount of water?

Answer

Approximately

43,255.2

ft-lb

Checkpoint 2.27 §2.5 source ↗

A water trough 12 m long has ends shaped like inverted isosceles triangles, with base 6 m and height 4 m. Find the force on one end of the trough if the trough is full of water.

Answer

156,800

Checkpoint 2.28 §2.5 source ↗

When the reservoir is at its average level, the surface of the water is about 50 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances?

Answer

Approximately 7,164,520,000 lb or 3,582,260 t

2-219 §2.5 source ↗

For the following exercises, find the work done.

How much work is done when a person lifts a $50$ lb box of comics onto a truck that is $3$ ft off the ground?

Answer

150

ft-lb

2-221 §2.5 source ↗

For the following exercises, find the work done.

Find the work done when you push a box along the floor $2$ m, when you apply a constant force of $F = 100 N .$

Answer

200 J

2-223 §2.5 source ↗

For the following exercises, find the work done.

What is the work done moving a particle from $x = 0$ to $x = 1$ m if the force acting on it is $F = 3 x^{2}$ N?

Answer

1

2-225 §2.5 source ↗

For the following exercises, find the mass of the one-dimensional object.

A car antenna that is $3$ ft long (starting at $x = 0)$ and has a density function of $ρ (x) = 3 x + 2$ lb/ft

Answer

\frac{39}{2}

2-227 §2.5 source ↗

For the following exercises, find the mass of the one-dimensional object.

A pencil that is $4$ in. long (starting at $x = 2)$ and has a density function of $ρ (x) = 5 / x$ oz/in.

Answer

ln (243)

2-229 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

An oversized hockey puck of radius $2$ in. with density function $ρ (x) =$ $x^{3} -$ $2 x +$ $5$

Answer

\frac{332 π}{15}

2-231 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A plate of radius $10$ in. with density function $ρ (x) =$ $1 +$ $cos (π x)$

Answer

100 π

2-233 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A disk of radius $5$ cm with density function $ρ (x) = \sqrt{3 x}$

Answer

20 π \sqrt{15}

2-235 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A spring has a natural length of $10$ cm. It takes $2$ J to stretch the spring to $15$ cm. How much work would it take to stretch the spring from $15$ cm to $20$ cm?

Answer

6

2-237 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A spring requires $5$ J to stretch the spring from $8$ cm to $12$ cm, and an additional $4$ J to stretch the spring from $12$ cm to $14$ cm. What is the natural length of the spring?

Answer

5

2-239 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A force of $F = 20 x - x^{3}$ N stretches a nonlinear spring by $x$ meters. What work is required to stretch the spring from $x = 0$ to $x = 2$ m?

Answer

36

2-241 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

For the cable in the preceding exercise, how much work is done to lift the cable $50$ ft?

Answer

18,750

ft-lb

2-243 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

[T] A pyramid of height $500$ ft has a square base $800$ ft by $800$ ft. Find the area $A$ at height $h .$ If the rock used to build the pyramid weighs approximately $w =$ $100 {lb/ft}^{3},$ how much work did it take to lift all the rock?

Answer

Weight $= \frac{32}{3} \times 10^{9} lb$

Work $=$ $\frac{4}{3} \times 10^{12} ft-lb$

2-245 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

[T] The force of gravity on a mass $m$ is $F =$ $− ((G M m) / x^{2})$ newtons. For a rocket of mass $m = 1000 kg,$ compute the work to lift the rocket from $x = 6400$ to $x = 6500$ km. State your answers with three significant figures. (Note: $G =$ $6.67 \times 10^{−11} {N m}^{2} / {kg}^{2}$ and $M =$ $6 \times 10^{24} kg .)$

Answer

9.71 \times 10^{8} N m

2-247 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

[T] A rectangular dam is $40$ ft high and $60$ ft wide. Assume the weight density of water is 62.5 lbs/ft³. Compute the total force $F$ on the dam when

a. the surface of the water is at the top of the dam and

b. the surface of the water is halfway down the dam.

Answer

3,000,000

lb, b.

750,000

lbs

2-249 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

Answer

23.25 π

million ft-lb

2-251 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A cylinder of depth $H$ and cross-sectional area $A$ stands full of water at density $ρ .$ Compute the work to pump all the water to the top.

Answer

\frac{A ρ H^{2}}{2}

2-253 §2.5 source ↗

For the following exercises, find the mass of the two-dimensional object that is centered at the origin.

A cone-shaped tank has a cross-sectional area that increases with its depth: $A =$ $(π r^{2} h^{2}) / H^{3} .$ Show that the work to empty it is half the work for a cylinder with the same height and base.

Answer

Answers may vary

Checkpoint 2.29 §2.6 source ↗

Suppose four point masses are placed on a number line as follows:

\begin{matrix} m_{1} = 12 kg, placed at x_{1} = −4 m & m_{2} = 12 kg, placed at x_{2} = 4 m \\ m_{3} = 30 kg, placed at x_{3} = 2 m & m_{4} = 6 kg, placed at x_{4} = −6 m . \end{matrix}

Find the moment of the system with respect to the origin and find the center of mass of the system.

Answer

M = 24, \bar{x} = \frac{2}{5} m

Checkpoint 2.30 §2.6 source ↗

Suppose three point masses are placed on a number line as follows (assume coordinates are given in meters):

\begin{matrix} m_{1} = 5 kg, placed at (−2, −3), \\ m_{2} = 3 kg, placed at (2, 3), \\ m_{3} = 2 kg, placed at (−3, −2) . \end{matrix}

Find the center of mass of the system.

Answer

(−1, −1)

Checkpoint 2.31 §2.6 source ↗

Let R be the region bounded above by the graph of the function

f (x) = x^{2}

and below by the x-axis over the interval

[0, 2] .

Find the centroid of the region.

Answer

The centroid of the region is

(3 / 2, 6 / 5) .

Checkpoint 2.32 §2.6 source ↗

Let R be the region bounded above by the graph of the function

f (x) = 6 - x^{2}

and below by the graph of the function

g (x) = 3 - 2 x .

Find the centroid of the region.

Answer

The centroid of the region is

(1, 13 / 5) .

Checkpoint 2.33 §2.6 source ↗

Let R be the region bounded above by the graph of the function

f (x) = 1 - x^{2}

and below by x-axis. Find the centroid of the region.

Answer

The centroid of the region is

(0, 2 / 5) .

Checkpoint 2.34 §2.6 source ↗

Let R be a circle of radius 1 centered at

(3, 0) .

Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y-axis.

Answer

6 π^{2}

units³

2-255 §2.6 source ↗

For the following exercises, calculate the center of mass for the collection of masses given.

$m_{1} = 1$ at $x_{1} = −1$ and $m_{2} = 3$ at $x_{2} = 2$

Answer

\frac{5}{4}

2-257 §2.6 source ↗

For the following exercises, calculate the center of mass for the collection of masses given.

Unit masses at $(x, y) =$ $(1, 0),$ $(0, 1),$ $(1, 1)$

Answer

(\frac{2}{3}, \frac{2}{3})

2-259 §2.6 source ↗

For the following exercises, calculate the center of mass for the collection of masses given.

$m_{1} = 1$ at $(1, 0)$ and $m_{2} = 3$ at $(2, 2)$

Answer

(\frac{7}{4}, \frac{3}{2})

2-261 §2.6 source ↗

For the following exercises, compute the center of mass $\bar{x} .$

$ρ = x^{2}$ for $x \in (0, L)$

Answer

\frac{3 L}{4}

2-263 §2.6 source ↗

For the following exercises, compute the center of mass $\bar{x} .$

$ρ = sin x$ for $x \in (0, π)$

Answer

\frac{π}{2}

2-265 §2.6 source ↗

For the following exercises, compute the center of mass $\bar{x} .$

$ρ = e^{x}$ for $x \in (0, 2)$

Answer

\frac{e^{2} + 1}{e^{2} - 1}

2-267 §2.6 source ↗

For the following exercises, compute the center of mass $\bar{x} .$

$ρ = x sin x$ for $x \in (0, π)$

Answer

\frac{π^{2} - 4}{π}

2-269 §2.6 source ↗

For the following exercises, compute the center of mass $\bar{x} .$

$ρ = ln x$ for $x \in (1, e)$

Answer

\frac{1}{4} (1 + e^{2})

2-271 §2.6 source ↗

For the following exercises, compute the center of mass $(\bar{x}, \bar{y}) .$ Use symmetry to help locate the center of mass whenever possible.

$ρ = 3$ in the triangle with vertices $(0, 0),$ $(a, 0),$ and $(0, b)$

Answer

(\frac{a}{3}, \frac{b}{3})

2-273 §2.6 source ↗

For the following exercises, use a calculator to draw the region, then compute the center of mass $(\bar{x}, \bar{y}) .$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y = cos (2 x),$ $x = - \frac{π}{4},$ and $x = \frac{π}{4}$

Answer

(0, \frac{π}{8})

2-275 §2.6 source ↗

For the following exercises, use a calculator to draw the region, then compute the center of mass $(\bar{x}, \bar{y}) .$ Use symmetry to help locate the center of mass whenever possible.

[T] The region between $y = \frac{5}{4} x^{2}$ and $y = 5$

Answer

(0, 3)

2-277 §2.6 source ↗

For the following exercises, use a calculator to draw the region, then compute the center of mass $(\bar{x}, \bar{y}) .$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y = 0,$ $\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1$

Answer

(0, \frac{4}{π})

2-279 §2.6 source ↗

For the following exercises, use a calculator to draw the region, then compute the center of mass $(\bar{x}, \bar{y}) .$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y = x^{2}$ and $y = x^{4}$ in the first quadrant

Answer

(\frac{5}{8}, \frac{1}{3})

2-281 §2.6 source ↗

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating $y = m x$ around the $y$ -axis between $x = 0$ and $x = 1$

Answer

\frac{2 m π}{3}

2-283 §2.6 source ↗

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

A general cylinder created by rotating a rectangle with vertices $(0, 0),$ $(a, 0),$ $(0, b),$ and $(a, b)$ around the $y$ -axis. Does your answer agree with the volume of a cylinder?

Answer

π a^{2} b

2-285 §2.6 source ↗

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $(\bar{x}, \bar{y})$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Quarter-circle: $y = \sqrt{1 - x^{2}},$ $y = 0,$ and $x = 0$

Answer

(\frac{4}{3 π}, \frac{4}{3 π})

2-287 §2.6 source ↗

[T] Lens: $y = x^{2}$ and $y = x$

Answer

(\frac{1}{2}, \frac{2}{5})

2-289 §2.6 source ↗

[T] Half-ring: $y^{2} + x^{2} = 1,$ $y^{2} + x^{2} = 4,$ and $y = 0$

Answer

(0, \frac{28}{9 π})

2-291 §2.6 source ↗

Find the generalized center of mass between $y = a^{2} - x^{2},$ $x = 0,$ and $y = 0 .$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Answer

Center of mass:

(\frac{3 a}{8}, \frac{2 a^{2}}{5}),

volume:

\frac{π a^{4}}{2}

2-293 §2.6 source ↗

Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius $a$ is positioned with the left end of the circle at $x = b,$ $b > 0,$ and is rotated around the y-axis.

This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.

Answer

Volume:

2 π^{2} a^{2} (b + a)

Checkpoint 2.35 §2.7 source ↗

Calculate the following derivatives:

a. $\frac{d}{d x} ln (2 x^{2} + x)$

b. $\frac{d}{d x} {(ln (x^{3}))}^{2}$

Answer

a. $\frac{d}{d x} ln (2 x^{2} + x) =$ $\frac{4 x + 1}{2 x^{2} + x}$

b. $\frac{d}{d x} {(ln (x^{3}))}^{2} =$ $\frac{6 ln (x^{3})}{x}$

Checkpoint 2.36 §2.7 source ↗

Calculate the integral

\int \frac{x^{2}}{x^{3} + 6} d x .

Answer

\int \frac{x^{2}}{x^{3} + 6} d x =

\frac{1}{3} ln | x^{3} +

6 | +

C

Checkpoint 2.37 §2.7 source ↗

Use properties of logarithms to simplify the following expression into a single logarithm:

ln 8 - ln 2 - ln (\frac{1}{4}) .

Answer

4 ln 2

Checkpoint 2.38 §2.7 source ↗

Evaluate the following derivatives:

a. $\frac{d}{d x} (\frac{e^{x^{2}}}{e^{5 x}})$

b. $\frac{d}{d t} {(e^{2 t})}^{3}$

Answer

a. $\frac{d}{d x} (\frac{e^{x^{2}}}{e^{5 x}}) =$ $e^{x^{2} - 5 x} (2 x - 5)$

b. $\frac{d}{d t} {(e^{2 t})}^{3} =$ $6 e^{6 t}$

Checkpoint 2.39 §2.7 source ↗

Evaluate the following integral:

\int \frac{4}{e^{3 x}} d x .

Answer

\int \frac{4}{e^{3 x}} d x =

-

\frac{4}{3} e^{−3 x} +

C

Checkpoint 2.40 §2.7 source ↗

Evaluate the following derivatives:

a. $\frac{d}{d t} 4^{t^{4}}$

b. $\frac{d}{d x} {log}_{3} (\sqrt{x^{2} + 1})$

Answer

a. $\frac{d}{d t} 4^{t^{4}} =$ $4^{t^{4}} (ln 4) (4 t^{3})$

b. $\frac{d}{d x} {log}_{3} (\sqrt{x^{2} + 1}) =$ $\frac{x}{(ln 3) (x^{2} + 1)}$

Checkpoint 2.41 §2.7 source ↗

Evaluate the following integral:

\int x^{2} 2^{x^{3}} d x .

Answer

\int x^{2} 2^{x^{3}} d x =

\frac{1}{3 ln 2} 2^{x^{3}} +

C

2-295 §2.7 source ↗

For the following exercises, find the derivative $\frac{d y}{d x} .$

$y = ln (2 x)$

Answer

\frac{1}{x}

2-297 §2.7 source ↗

For the following exercises, find the derivative $\frac{d y}{d x} .$

$y = \frac{1}{ln x}$

Answer

- \frac{1}{x {(ln x)}^{2}}

2-299 §2.7 source ↗

For the following exercises, find the indefinite integral.

$\int \frac{d x}{1 + x}$

Answer

ln (x + 1) + C

2-301 §2.7 source ↗

For the following exercises, find the derivative $d y / d x .$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y = x ln (x)$

Answer

ln (x) + 1

2-303 §2.7 source ↗

For the following exercises, find the derivative $d y / d x .$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y = ln (sin x)$

Answer

cot (x)

2-305 §2.7 source ↗

For the following exercises, find the derivative $d y / d x .$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y = 7 ln (4 x)$

Answer

\frac{7}{x}

2-307 §2.7 source ↗

For the following exercises, find the derivative $d y / d x .$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y = ln (tan x)$

Answer

csc (x) sec x

2-309 §2.7 source ↗

For the following exercises, find the derivative $d y / d x .$ (You can use a calculator to plot the function and the derivative to confirm that it is correct.)

[T] $y = ln ({cos}^{2} x)$

Answer

−2 tan x

2-311 §2.7 source ↗

For the following exercises, find the definite or indefinite integral.

$\int_{0}^{1} \frac{d t}{3 + 2 t}$

Answer

\frac{1}{2} ln (\frac{5}{3})

2-313 §2.7 source ↗

For the following exercises, find the definite or indefinite integral.

$\int_{0}^{2} \frac{x^{3} d x}{x^{2} + 1}$

Answer

2 - \frac{1}{2} ln (5)

2-315 §2.7 source ↗

For the following exercises, find the definite or indefinite integral.

$\int_{2}^{e} \frac{d x}{{x (ln x)}^{2}}$

Answer

\frac{1}{ln (2)} - 1

2-317 §2.7 source ↗

For the following exercises, find the definite or indefinite integral.

$\int_{0}^{π / 4} tan x d x$

Answer

\frac{1}{2} ln (2)

2-319 §2.7 source ↗

For the following exercises, find the definite or indefinite integral.

$\int \frac{{(ln x)}^{2} d x}{x}$

Answer

\frac{1}{3} {(ln x)}^{3}

2-321 §2.7 source ↗

For the following exercises, compute $d y / d x$ by differentiating $ln y .$

$y = \sqrt{x^{2} + 1} \sqrt{x^{2} - 1}$

Answer

\frac{2 x^{3}}{\sqrt{x^{2} + 1} \sqrt{x^{2} - 1}}

2-323 §2.7 source ↗

For the following exercises, compute $d y / d x$ by differentiating $ln y .$

$y = x^{−1 / x}$

Answer

x^{−2 - (1 / x)} (ln x - 1)

2-325 §2.7 source ↗

For the following exercises, compute $d y / d x$ by differentiating $ln y .$

$y = x^{e}$

Answer

e x^{e - 1}

2-327 §2.7 source ↗

For the following exercises, compute $d y / d x$ by differentiating $ln y .$

$y = \sqrt{x} \sqrt[3]{x} \sqrt[6]{x}$

Answer

1

2-329 §2.7 source ↗

For the following exercises, compute $d y / d x$ by differentiating $ln y .$

$y = e^{− ln x}$

Answer

- \frac{1}{x^{2}}

2-331 §2.7 source ↗

For the following exercises, evaluate by any method.

$\int_{1}^{e^{π}} \frac{d x}{x} +$ $\int_{−2}^{−1} \frac{d x}{x}$

Answer

π - ln (2)

2-333 §2.7 source ↗

For the following exercises, evaluate by any method.

$\frac{d}{d x} \int_{x}^{x^{2}} \frac{d t}{t}$

Answer

\frac{1}{x}

2-335 §2.7 source ↗

For the following exercises, use the function $ln x .$ If you are unable to find intersection points analytically, use a calculator.

Find the area of the region enclosed by $x = 1$ and $y = 5$ above $y = ln x .$

Answer

e^{5} - 6 {units}^{2}

2-337 §2.7 source ↗

For the following exercises, use the function $ln x .$ If you are unable to find intersection points analytically, use a calculator.

Find the area between $ln x$ and the x-axis from $x = 1 to x = 2 .$

Answer

ln (4) -

1 {units}^{2}

2-339 §2.7 source ↗

For the following exercises, use the function $ln x .$ If you are unable to find intersection points analytically, use a calculator.

[T] Find the surface area of the shape created when rotating the curve in the previous exercise from $x = 1$ to $x = 2$ around the x-axis.

Answer

2.8656

2-341 §2.7 source ↗

If you are unable to find intersection points analytically in the following exercises, use a calculator.

[T] Find the arc length of $y = 1 / x$ from $x = 1 to x = 4 .$

Answer

3.1502

Checkpoint 2.42 §2.8 source ↗

Consider a population of bacteria that grows according to the function

f (t) =

500 e^{0.05 t},

where

t

is measured in minutes. How many bacteria are present in the population after 4 hours? When does the population reach

100

million bacteria?

Answer

There are

81,377,396

bacteria in the population after

4

hours. The population reaches

100

million bacteria after

244.12

minutes.

Checkpoint 2.43 §2.8 source ↗

Suppose instead of investing at age

25

, the student waits until age

35 .

How much would she have to invest at

5 % ?

6 % ?

Answer

5 %

interest, she must invest

$ 223,130.16 .

6 %

interest, she must invest

$ 165,298.89 .

Checkpoint 2.44 §2.8 source ↗

Suppose it takes

9

months for the fish population in Example 2.44 to reach

1000

fish. Under these circumstances, how long do the owner’s friends have to wait?

Answer

38.90

months

Checkpoint 2.45 §2.8 source ↗

Suppose the room is warmer

(75 ° F)

and, after

2

minutes, the coffee has cooled only to

185 ° F .

When is the coffee first cool enough to serve? When is the coffee be too cold to serve? Round answers to the nearest half minute.

Answer

The coffee is first cool enough to serve about

3.5

minutes after it is poured. The coffee is too cold to serve about

7

minutes after it is poured.

Checkpoint 2.46 §2.8 source ↗

If we have

100

g of

carbon- 14,

how much is left after

500

years? If an artifact that originally contained

100

g of carbon now contains

20 g

of carbon, how old is it? Round the answer to the nearest hundred years.

Answer

A total of

94.13

g of carbon remains. The artifact is approximately

13,300

years old.

2-349 §2.8 source ↗

True or False? If true, prove it. If false, find the true answer.

If you invest $$ 500,$ an annual rate of interest of $3 %$ yields more money in the first year than a $2.5 %$ continuous rate of interest.

Answer

True

2-351 §2.8 source ↗

True or False? If true, prove it. If false, find the true answer.

If given a half-life of t years, the constant $k$ for $y = e^{k t}$ is calculated by $k =$ $ln (1 / 2) / t .$

Answer

True;

k = \frac{ln (2)}{t}

2-353 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

If bacteria increase by a factor of $10$ in $10$ hours, how many hours does it take to increase by $100 ?$

Answer

20

hours

2-355 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

If a relic contains $90 %$ as much radiocarbon as new material, can it have come from the time of Christ (approximately $2000$ years ago)? Note that the half-life of radiocarbon is $5730$ years.

Answer

No. The relic is approximately

871

years old.

2-357 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

The populations of New York and Los Angeles are growing at $1 %$ and $1.4 %$ a year, respectively. Starting from $8$ million (New York) and $6$ million (Los Angeles), when are the populations equal? Round your answer to a whole number of years.

Answer

73

years

2-359 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

The effect of advertising decays exponentially. If $40 %$ of the population remembers a new product after $3$ days, how long will $20 %$ remember it?

Answer

5

days

6

hours

27

minutes

2-361 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

If $y = 100$ at $t = 4$ and $y = 10$ at $t = 8,$ when does $y = 1 ?$

Answer

12

2-363 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

What continuous interest rate has the same yield as an annual rate of $9 % ?$

Answer

8.618 %

2-365 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

You are trying to save $$ 50,000$ in $20$ years for college tuition for your child. If interest is a continuous $10 %,$ how much do you need to invest initially?

Answer

$ 6766.76

2-367 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

You are trying to thaw some vegetables that are at a temperature of $1 ° F .$ To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $44 ° F .$ You check on your vegetables $2$ hours after putting them in the refrigerator to find that they are now $12 ° F .$ Plot the resulting temperature curve and use it to determine when the vegetables reach $33 ° F .$

Answer

9

hours

13

minutes

2-369 §2.8 source ↗

For the following exercises, use $y = y_{0} e^{k t} .$

The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $24,000$ years. If $1$ barrel containing $10 kg$ of plutonium-239 is sealed, how many years must pass until only $10 g$ of plutonium-239 is left?

Answer

239,179

years

2-371 §2.8 source ↗

For the next set of exercises, use the following table, which features the world population by decade.

Years since 1950	Population (millions)
$0$	$2,556$
$10$	$3,039$
$20$	$3,706$
$30$	$4,453$
$40$	$5,279$
$50$	$6,083$
$60$	$6,849$

Source: http://www.factmonster.com/ipka/A0762181.html.

[T] Find and graph the derivative $y^{'}$ of your equation. Where is it increasing and what is the meaning of this increase?

Answer

P' (t) =

43 e^{0.01604 t} .

The population is always increasing.

2-373 §2.8 source ↗

For the next set of exercises, use the following table, which features the world population by decade.

Years since 1950	Population (millions)
$0$	$2,556$
$10$	$3,039$
$20$	$3,706$
$30$	$4,453$
$40$	$5,279$
$50$	$6,083$
$60$	$6,849$

Source: http://www.factmonster.com/ipka/A0762181.html.

[T] Find the predicted date when the population reaches $10$ billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.

Answer

The population reaches

10

billion people in

2032 .

2-375 §2.8 source ↗

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century.

Years since 1850	Population (thousands)
$0$	$21.00$
$10$	$56.80$
$20$	$149.5$
$30$	$234.0$

Source: http://www.sfgenealogy.com/sf/history/hgpop.htm.

[T] Find and graph the derivative $y^{'}$ of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?

Answer

P' (t) =

2.259 e^{0.06407 t} .

The population is always increasing.

Checkpoint 2.47 §2.9 source ↗

Evaluate the following derivatives:

a. $\frac{d}{d x} (tanh (x^{2} + 3 x))$

b. $\frac{d}{d x} (\frac{1}{{(sinh x)}^{2}})$

Answer

a. $\frac{d}{d x} (tanh (x^{2} + 3 x)) =$ $({sech}^{2} (x^{2} + 3 x)) (2 x + 3)$

b. $\frac{d}{d x} (\frac{1}{{(sinh x)}^{2}}) =$ $\frac{d}{d x} {(sinh x)}^{−2} =$ $−2 {(sinh x)}^{−3} cosh x$

Checkpoint 2.48 §2.9 source ↗

Evaluate the following integrals:

a. $\int {sinh}^{3} x cosh x d x$

b. $\int {sech}^{2} (3 x) d x$

Answer

a. $\int {sinh}^{3} x cosh x d x =$ $\frac{{sinh}^{4} x}{4} +$ $C$

b. $\int {sech}^{2} (3 x) d x =$ $\frac{tanh (3 x)}{3} +$ $C$

Checkpoint 2.49 §2.9 source ↗

Evaluate the following derivatives:

a. $\frac{d}{d x} ({cosh}^{−1} (3 x))$

b. $\frac{d}{d x} {({coth}^{−1} x)}^{3}$

Answer

a. $\frac{d}{d x} ({cosh}^{−1} (3 x)) =$ $\frac{3}{\sqrt{9 x^{2} - 1}}$

b. $\frac{d}{d x} {({coth}^{−1} x)}^{3} =$ $\frac{3 {({coth}^{−1} x)}^{2}}{1 - x^{2}}$

Checkpoint 2.50 §2.9 source ↗

Evaluate the following integrals:

a. $\int \frac{1}{\sqrt{x^{2} - 4}} d x,$ $x >$ $2$

b. $\int \frac{1}{\sqrt{1 - e^{2 x}}} d x$

Answer

a. $\int \frac{1}{\sqrt{x^{2} - 4}} d x =$ ${cosh}^{−1} (\frac{x}{2}) +$ $C$

b. $\int \frac{1}{\sqrt{1 - e^{2 x}}} d x =$ $− {sech}^{−1} (e^{x}) +$ $C$

Checkpoint 2.51 §2.9 source ↗

Assume a hanging cable has the shape

15 cosh (x / 15)

for

−20 \leq x \leq 20 .

Determine the length of the cable (in feet).

Answer

52.95 ft

2-377 §2.9 source ↗

[T] Find expressions for

cosh x + sinh x

and

cosh x -

sinh x .

Use a calculator to graph these functions and ensure your expression is correct.

Answer

e^{x} and e^{− x}

2-379 §2.9 source ↗

Show that

cosh (x)

and

sinh (x)

satisfy

y^{″} = y .

Answer

Answers may vary

2-381 §2.9 source ↗

Derive

{cosh}^{2} (x) +

{sinh}^{2} (x) =

cosh (2 x)

from the definition.

Answer

Answers may vary

2-383 §2.9 source ↗

Prove

sinh (x + y) =

sinh (x) cosh (y) +

cosh (x) sinh (y)

by changing the expression to exponentials.

Answer

Answers may vary

2-385 §2.9 source ↗

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

[T] $cosh (3 x + 1)$

Answer

3 sinh (3 x + 1)

2-387 §2.9 source ↗

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

[T] $\frac{1}{cosh (x)}$

Answer

− tanh (x) sech (x)

2-389 §2.9 source ↗

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

[T] ${cosh}^{2} (x) +$ ${sinh}^{2} (x)$

Answer

4 cosh (x) sinh (x)

2-391 §2.9 source ↗

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

[T] $tanh (\sqrt{x^{2} + 1})$

Answer

\frac{x {sech}^{2} (\sqrt{x^{2} + 1})}{\sqrt{x^{2} + 1}}

2-393 §2.9 source ↗

For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

[T] ${sinh}^{6} (x)$

Answer

6 {sinh}^{5} (x) cosh (x)

2-395 §2.9 source ↗

For the following exercises, find the antiderivatives for the given functions.

$cosh (2 x + 1)$

Answer

\frac{1}{2} sinh (2 x + 1) +

C

2-397 §2.9 source ↗

For the following exercises, find the antiderivatives for the given functions.

$x cosh (x^{2})$

Answer

\frac{1}{2} sinh (x^{2}) +

C

2-399 §2.9 source ↗

For the following exercises, find the antiderivatives for the given functions.

${cosh}^{2} (x) sinh (x)$

Answer

\frac{1}{3} {cosh}^{3} (x) +

C

2-401 §2.9 source ↗

For the following exercises, find the antiderivatives for the given functions.

$\frac{sinh (x)}{1 + cosh (x)}$

Answer

ln (1 + cosh (x)) +

C

2-403 §2.9 source ↗

For the following exercises, find the antiderivatives for the given functions.

$cosh (x) +$ $sinh (x)$

Answer

cosh (x) +

sinh (x) +

C

2-405 §2.9 source ↗

For the following exercises, find the derivatives for the functions.

${tanh}^{−1} (4 x)$

Answer

\frac{4}{1 - 16 x^{2}}

2-407 §2.9 source ↗

For the following exercises, find the derivatives for the functions.

${sinh}^{−1} (cosh (x))$

Answer

\frac{sinh (x)}{\sqrt{{cosh}^{2} (x) + 1}}

2-409 §2.9 source ↗

For the following exercises, find the derivatives for the functions.

${tanh}^{−1} (cos (x))$

Answer

− csc (x)

2-411 §2.9 source ↗

For the following exercises, find the derivatives for the functions.

$ln ({tanh}^{−1} (x))$

Answer

-

\frac{1}{(x^{2} - 1) {tanh}^{−1} (x)}

2-413 §2.9 source ↗

For the following exercises, find the antiderivatives for the functions.

$\int \frac{d x}{a^{2} - x^{2}}$

Answer

\frac{1}{a} {tanh}^{−1} (\frac{x}{a}) +

C

2-415 §2.9 source ↗

For the following exercises, find the antiderivatives for the functions.

$\int \frac{x d x}{\sqrt{x^{2} + 1}}$

Answer

\sqrt{x^{2} + 1} + C

2-417 §2.9 source ↗

For the following exercises, find the antiderivatives for the functions.

$\int \frac{e^{x}}{\sqrt{e^{2 x} - 1}} d x$

Answer

{cosh}^{−1} (e^{x}) +

C

2-419 §2.9 source ↗

For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation $d v / d t = g - v^{2} .$

Show that $v (t) =$ $\sqrt{g} tanh ((\sqrt{g}) t)$ satisfies this equation.

Answer

Answers may vary

2-421 §2.9 source ↗

For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation $d v / d t = g - v^{2} .$

[T] Estimate how far a body has fallen in $12$ seconds by finding the area underneath the curve of $v (t) .$

Answer

37.30

2-423 §2.9 source ↗

For the following exercises, use this scenario: A cable hanging under its own weight has a slope $S = d y / d x$ that satisfies $d S / d x =$ $c \sqrt{1 + S^{2}} .$ The constant $c$ is the ratio of cable density to tension.

Integrate $d y / d x =$ $sinh (c x)$ to find the cable height $y (x)$ if $y (0) = 1 / c .$

Answer

y =

\frac{1}{c} cosh (c x)

2-425 §2.9 source ↗

For the following exercises, solve each problem.

[T] A chain hangs from two posts $2$ m apart to form a catenary described by the equation $y =$ $2 cosh (x / 2) -$ $1 .$ Find the slope of the catenary at the left fence post.

Answer

−0.521095

2-427 §2.9 source ↗

For the following exercises, solve each problem.

[T] A high-voltage power line is a catenary described by $y =$ $10 cosh (x / 10) .$ Find the ratio of the area under the catenary to its arc length. What do you notice?

Answer

10

2-435 §2.R source ↗

True or False? Justify your answer with a proof or a counterexample.

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

Answer

False

2-437 §2.R source ↗

True or False? Justify your answer with a proof or a counterexample.

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

Answer

False

2-439 §2.R source ↗

For the following exercises, use the requested method to determine the volume of the solid.

The volume that has a base of the ellipse $x^{2} / 4 + y^{2} / 9 = 1$ and cross-sections of an equilateral triangle perpendicular to the $y -axis .$ Use the method of slicing.

Answer

16 \sqrt{3} {units}^{3}

2-441 §2.R source ↗

For the following exercises, use the requested method to determine the volume of the solid.

$x = y^{2}$ and $x = 3 y$ rotated around the y-axis using the washer method

Answer

\frac{162 π}{5}

2-443 §2.R source ↗

For the following exercises, find

$y =$ $x^{3},$ $x =$ $0,$ $y =$ $0,$ $and x =$ $2$

Answer

4,

\frac{128 π}{7},

\frac{64 π}{5}

2-445 §2.R source ↗

For the following exercises, find

[T] $y =$ $ln (x) +$ $2 and y =$ $x$

Answer

1.949,

21.952,

17.099

2-447 §2.R source ↗

For the following exercises, find

$y = 5 + x,$ $y = x^{2},$ $x = 0,$ and $x = 1$

Answer

\frac{31}{6},

\frac{452 π}{15},

\frac{31 π}{6}

2-449 §2.R source ↗

For the following exercises, find

Find the mass of $ρ = \frac{1}{x^{2} + 1}$ on a disk centered at the origin with radius $4 .$

Answer

π ln 17

2-451 §2.R source ↗

For the following exercises, find

Find the mass and the center of mass of $ρ = 1$ on the region bounded by $y = x^{5}$ and $y = \sqrt{x} .$

Answer

Mass:

\frac{1}{2},

center of mass:

(\frac{18}{35}, \frac{9}{22})

2-453 §2.R source ↗

For the following exercises, find the requested arc lengths.

The length of $y$ for $x = 3 - \sqrt{y}$ from $y = 0$ to $y = 4$

Answer

\sqrt{17} +

\frac{1}{8} ln (33 + 8 \sqrt{17})

2-455 §2.R source ↗

For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.

The loudspeaker created by revolving $y = 1 / x$ from $x = 1$ to $x = 4$ around the x-axis.

Answer

Volume:

\frac{3 π}{4},

surface area:

π (\sqrt{2} - {sinh}^{−1} (1) + {sinh}^{−1} (16) - \frac{\sqrt{257}}{16})

2-457 §2.R source ↗

For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and $45 ° F$ outside and the temperature of the body is $78 ° F .$ You know the cooling constant is $k =$ $0.00824 ° F/min .$ When did the victim die, assuming that a human’s temperature is $98 ° F$ ?

Answer

11:02 a.m.

2-459 §2.R source ↗

For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings.

Find the volume of the catenoid $y = cosh (x)$ from $x = −1 to x = 1$ that is created by rotating this curve around the $x -axis,$ as shown here.

This figure is an image of a catenoid. It has been formed by rotating a catenary curve about a vertical axis.

Answer

π (1 + sinh (1) cosh (1))

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Problems and answers are adapted from Calculus Volume 2 by Gilbert Strang and Edwin “Jed” Herman, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.