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Thomas Calculus

George B. Thomas, Jr.

Chapter 6

Applications of Definite Integrals

Educators


Problem 1

The solid lies between planes perpendicular to the $x$ -axis at $x=0$
and $x=4 .$ The cross-sections perpendicular to the axis on the interval $0 \leq x \leq 4$ are squares whose diagonals run from the parabola $y=-\sqrt{x}$ to the parabola $y=\sqrt{x}$ .

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Problem 2

The solid lies between planes perpendicular to the $x$ -axis at
$x=-1$ and $x=1 .$ The cross-sections perpendicular to the
$x$ -axis are circular disks whose diameters run from the parabola
$y=x^{2}$ to the parabola $y=2-x^{2}$

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Problem 3

The solid lies between planes perpendicular to the $x$ -axis at
$x=-1$ and $x=1 .$ The cross-sections perpendicular to the
$x$ -axis between these planes are squares whose bases run from the
semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$

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Problem 4

The solid lies between planes perpendicular to the $x$ -axis at
$x=-1$ and $x=1 .$ The cross-sections perpendicular to the $x$ -axis
between these planes are squares whose diagonals run from the
semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$

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Problem 5

The base of a solid is the region between the curve $y=2 \sqrt{\sin x}$
and the interval $[0, \pi]$ on the $x$ -axis. The cross-sections perpendicular to the $x$ -axis are
a. equilateral triangles with bases running from the $x$ -axis to the
curve as shown in the accompanying figure.
b. squares with bases running from the $x$ -axis to the curve.

Kelsay S.
Numerade Educator

Problem 6

The solid lies between planes perpendicular to the $x$ -axis at
$x=-\pi / 3$ and $x=\pi / 3 .$ The cross-sections perpendicular to
the $x$ -axis are
\begin{equation}
\begin{array}{l}{\text { a. circular disks with diameters running from the curve }} \\ {y=\tan x \text { to the curve } y=\sec x .} \\ {\text { b. squares whose bases run from the curve } y=\tan x \text { to the }} \\ {\text { curve } y=\sec x .}\end{array}
\end{equation}

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Problem 7

The solid lies between planes perpendicular to the $x$ -axis at
$x=-\pi / 3$ and $x=\pi / 3 .$ The cross-sections perpendicular to
the $x$ -axis are
\begin{equation}
\begin{array}{l}{\text { a. rectangles of height } 10 \text { . }} \\ {\text { b. rectangles of perimeter } 20 \text { . }}\end{array}
\end{equation}

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Problem 8

The base of a solid is the region bounded by the graphs of
$y=\sqrt{x}$ and $y=x / 2 .$ The cross-sections perpendicular to the
$x$ -axis are
\begin{equation}
\begin{array}{l}{\text { a. isosceles triangles of height } 6 \text { . }} \\ {\text { b. semicircles with diameters running across the base of the solid. }}\end{array}
\end{equation}

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Problem 9

The solid lies between planes perpendicular to the $y$ -axis at $y=0$
and $y=2 .$ The cross-sections perpendicular to the $y$ -axis are cir-
cular disks with diameters running from the $y$ -axis to the parabola
$x=\sqrt{5} y^{2}$

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Problem 10

The base of the solid is the disk $x^{2}+y^{2} \leq 1 .$ The cross-sections
by planes perpendicular to the $y$ -axis between $y=-1$ and $y=1$
are isosceles right triangles with one leg in the disk.

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Problem 11

Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.)

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Problem 12

Find the volume of the given pyramid, which has a square base of
area 9 and height $5 .$

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Problem 13

A twisted solid A square of side length $s$ lies in a plane perpendicular to a line $L .$ One vertex of the square lies on $L .$ As this square moves a distance $h$ along $L$ , the square turns one revolution about $L$ to generate a corkscrew like column with square cross-sections.
\begin{equation}
\begin{array}{l}{\text { a. Find the volume of the column. }} \\ {\text { b. What will the volume be if the square turns twice instead of }} \\ {\text { once? Give reasons for your answer. }}\end{array}
\end{equation}

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Problem 14

Cavalieri's principle A solid lies between planes perpendicular
to the $x$ -axis at $x=0$ and $x=12$ . The cross-sections by planes
perpendicular to the $x$ -axis are circular disks whose diameters run from the line $y=x / 2$ to the line $y=x$ as shown in the accompanying figure. Explain why the solid has the same volume as a
right circular cone with base radius 3 and height $12 .$

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Problem 15

In Exercises $15-18$ , find the volume of the solid generated by revolving the shaded region about the given axis.

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Problem 16

Find the volume of the solid generated by revolving the shaded region about the given axis.

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Problem 17

Find the volume of the solid generated by revolving the shaded region about the given axis.

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Problem 18

Find the volume of the solid generated by revolving the shaded region about the given axis.

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Problem 19

Find the volumes of the solids generated by revolving the regions
bounded by the lines and curves in Exercises $19-28$ about the $x$ -axis.
\begin{equation}
y=x^{2}, \quad y=0, \quad x=2
\end{equation}

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Problem 20

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the $x$ -axis.
\begin{equation}
y=x^{3}, \quad y=0, \quad x=2
\end{equation}

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Problem 21

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the $x$ -axis.
\begin{equation}
y=\sqrt{9-x^{2}}, \quad y=0
\end{equation}

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Problem 22

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the $x$ -axis.
\begin{equation}
y=x-x^{2}, \quad y=0
\end{equation}

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Problem 23

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the $x$ -axis.
\begin{equation}
y=\sqrt{\cos x}, \quad 0 \leq x \leq \pi / 2, \quad y=0, \quad x=0
\end{equation}

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Problem 24

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the $x$ -axis.
\begin{equation}
y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4
\end{equation}

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Problem 25

In Exercises 25 and $26,$ find the volume of the solid generated by revolving the region about the given line.
\begin{equation}
\begin{array}{l}{\text { The region in the first quadrant bounded above by the line }} \\ {y=\sqrt{2}, \text { below by the curve } y=\sec x \tan x, \text { and on the left by }} \\ {\text { the } y \text { -axis, about the line } y=\sqrt{2}}\end{array}
\end{equation}

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Problem 26

Find the volume of the solid generated by revolving the region about the given line.
\begin{equation}
\begin{array}{l}{\text { The region in the first quadrant bounded above by the line } y=2} \\ {\text { below by the curve } y=2 \sin x, 0 \leq x \leq \pi / 2, \text { and on the left by }} \\ {\text { the } y \text { -axis, about the line } y=2}\end{array}
\end{equation}

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Problem 27

\begin{equation}
\begin{array}{l}{\text { Find the volumes of the solids generated by revolving the regions }} \\ {\text { bounded by the lines and curves in Exercises } 27-32 \text { about the } y \text { -axis. }}\end{array}
\end{equation}
The region enclosed by $x=\sqrt{5} y^{2}, \quad x=0, \quad y=-1, \quad y=1$

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Problem 28

\begin{equation}\begin{array}{l}{\text { Find the volumes of the solids generated by revolving the regions }} \\ {\text { bounded by the lines and curves in Exercises } 27-32 \text { about the } y \text { -axis. }}\end{array}\end{equation}
The region enclosed by $x=y^{3 / 2}, \quad x=0, \quad y=2$

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Problem 29

\begin{equation}\begin{array}{l}{\text { Find the volumes of the solids generated by revolving the regions }} \\ {\text { bounded by the lines and curves in Exercises } 27-32 \text { about the } y \text { -axis. }}\end{array}\end{equation}
The region enclosed by $x=\sqrt{2 \sin 2 y}, \quad 0 \leq y \leq \pi / 2, \quad x=0$

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Problem 30

\begin{equation}\begin{array}{l}{\text { Find the volumes of the solids generated by revolving the regions }} \\ {\text { bounded by the lines and curves in Exercises } 27-32 \text { about the } y \text { -axis. }}\end{array}\end{equation}
The region enclosed by $x=\sqrt{\cos (\pi y / 4)}, \quad-2 \leq y \leq 0$
$x=0$

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Problem 31

Find the volumes of the solids generated by revolving the regions
bounded by the lines and curves in Exercises $27-32$ about the $y$ -axis.
\begin{equation}
x=y^{1 / 3}, \quad x=y^{3}, \quad 0 \leq y \leq 1
\end{equation}

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Problem 32

Find the volumes of the solids generated by revolving the regions
bounded by the lines and curves in Exercises $27-32$ about the $y$ -axis.
\begin{equation}
x=\sqrt{2 y} /\left(y^{2}+1\right), \quad x=0, \quad y=1
\end{equation}

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Problem 33

Find the volumes of the solids generated by revolving the shaded regions in Exercises 33 and 34 about the indicated axes.
The $x$ -axis

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Problem 34

Find the volumes of the solids generated by revolving the shaded regions about the indicated axes.
The $y$ -axis

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Problem 35

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $35-40$ about the $x$ -axis.
\begin{equation}
y=x, \quad y=1, \quad x=0
\end{equation}

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Problem 36

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $35-40$ about the $x$ -axis.
\begin{equation}
y=2 \sqrt{x}, \quad y=2, \quad x=0
\end{equation}

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Problem 37

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $35-40$ about the $x$ -axis.
\begin{equation}
y=x^{2}+1, y=x+3
\end{equation}

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Problem 38

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $35-40$ about the $x$ -axis.
\begin{equation}
y=4-x^{2}, \quad y=2-x
\end{equation}

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Problem 39

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $35-40$ about the $x$ -axis.
\begin{equation}
y=\sec x, \quad y=\sqrt{2}, \quad-\pi / 4 \leq x \leq \pi / 4
\end{equation}

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Problem 40

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $35-40$ about the $x$ -axis.
\begin{equation}
y=\sec x, \quad y=\tan x, \quad x=0, \quad x=1
\end{equation}

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Problem 41

In Exercises $41-44,$ find the volume of the solid generated by revolving each region about the $y$ -axis.
\begin{equation}
\begin{array}{l}{\text { The region enclosed by the triangle with vertices }(1,0),(2,1),} \\ {\text { and }(1,1)}\end{array}
\end{equation}

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Problem 42

Find the volume of the solid generated by revolving each region about the $y$ -axis.
\begin{equation}
\begin{array}{l}{\text { The region enclosed by the triangle with vertices }(0,1),(1,0),} \\ {\text { and }(1,1)}\end{array}
\end{equation}

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Problem 43

Find the volume of the solid generated by revolving each region about the $y$ -axis.
\begin{equation}
\begin{array}{l}{\text { The region in the first quadrant bounded above by the parabola }} \\ {y=x^{2}, \text { below by the } x \text { -axis, and on the right by the line } x=2}\end{array}
\end{equation}

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Problem 44

Find the volume of the solid generated by revolving each region about the $y$ -axis.
\begin{equation}
\begin{array}{l}{\text { The region in the first quadrant bounded on the left by the circle }} \\ {x^{2}+y^{2}=3, \text { on the right by the line } x=\sqrt{3}, \text { and above by the }} \\ {\text { line } y=\sqrt{3}}\end{array}
\end{equation}

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Problem 45

In Exercises 45 and $46,$ find the volume of the solid generated by revolving each region about the given axis.
\begin{equation}
\begin{array}{l}{\text { The region in the first quadrant bounded above by the curve }} \\ {y=x^{2}, \text { below by the } x \text { -axis, and on the right by the line } x=1 \text { , }} \\ {\text { about the line } x=-1}\end{array}
\end{equation}

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Problem 46

Find the volume of the solid generated by revolving each region about the given axis.
\begin{equation}
\begin{array}{l}{\text { The region in the second quadrant bounded above by the curve }} \\ {y=-x^{3}, \text { below by the } x \text { -axis, and on the left by the line } x=-1} \\ {\text { about the line } x=-2}\end{array}
\end{equation}

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Problem 47

Find the volume of the solid generated by revolving the region bounded by $y=\sqrt{x}$ and the lines $y=2$ and $x=0$ about
\begin{equation}
\begin{array}{ll}{\text { a. the } x \text { -axis. }} & {\text { b. the } y \text { -axis. }} \\ {\text { c. the line } y=2 .} & {\text { d. the line } x=4}\end{array}
\end{equation}

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Problem 48

Find the volume of the solid generated by revolving the triangular region bounded by the lines $y=2 x, y=0,$ and $x=1$ about
\begin{equation}
{a. x =1}. \quad\quad \quad \text { b. the line } x =2
\end{equation}

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Problem 49

Find the volume of the solid generated by revolving the region bounded by the parabola $y=x^{2}$ and the line $y=1$ about
\begin{equation}
\begin{array}{ll}{\text { a. the line } y=1 .} & {\text { b. the line } y=2 \text { . }} \\ {\text { c. the line } y=-1}\end{array}
\end{equation}

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Problem 50

By integration, find the volume of the solid generated by revolving the triangular region with vertices $(0,0),(b, 0),(0, h)$ about
\begin{equation}
{ a.x } \text { -axis. } \quad \text { b. the } y
\end{equation}

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Problem 51

The volume of a torus The disk $x^{2}+y^{2} \leq a^{2}$ is revolved about the line $x=b(b>a)$ to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint:
$\int_{-a}^{a} \sqrt{a^{2}}-y^{2} d y=\pi a^{2} / 2,$ since it is the area of a semicircle of
radius $a$ .

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Problem 52

Volume of a bowl A bowl has a shape that can be generated by revolving the graph of $y=x^{2} / 2$ between $y=0$ and $y=5$ about the $y$ -axis.
\begin{equation}
\begin{array}{l}{\text { a. Find the volume of the bowl. }} \\ {\text { b. Related rates If we fill the bowl with water at a constant }} \\ {\text { rate of } 3 \text { cubic units per second, how fast will the water level }} \\ {\text { in the bowl be rising when the water is 4 units deep? }}\end{array}
\end{equation}

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Problem 53

Volume of a bowl
\begin{equation}
\begin{array}{l}{\text { a. A hemispherical bowl of radius } a \text { contains water to a depth } h .} \\ {\text { Find the volume of water in the bowl. }} \\ {\text { b. Related rates } \text { Water runs into a sunken concrete hemi- }} \\ {\text { spherical bowl of radius } 5 \mathrm{m} \text { at the rate of } 0.2 \mathrm{m}^{3} / \mathrm{sec} \text { . How }} \\ {\text { fast is the water level in the bowl rising when the water is }} \\ {4 \mathrm{m} \text { deep? }}\end{array}
\end{equation}

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Problem 54

Explain how you could estimate the volume of a solid of revolution by measuring the shadow cast on a table parallel to its axis of revolution by a light shining directly above it.

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Problem 55

Volume of a hemisphere Derive the formula $V=(2 / 3) \pi R^{3}$
for the volume of a hemisphere of radius $R$ by comparing its
cross-sections with the cross-sections of a solid right circular cylinder of radius $R$ and height $R$ from which a solid right circular cone of base radius $R$ and height $R$ has been removed, as suggested by the accompanying figure.

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Problem 56

Designing a plumb bob Having been asked to design a brass
plumb bob that will weigh in the neighborhood of 190 g, you
decide to shape it like the solid of revolution shown here. Find the
plumb bob's volume. If you specify a brass that weighs 8.5 $\mathrm{g} / \mathrm{cm}^{3}$ ,
how much will the plumb bob weigh (to the nearest gram)?

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Problem 57

Designing a wok You are designing a wok frying pan that will
be shaped like a spherical bowl with handles. A bit of experimentation at home persuades you that you can get one that holds about 3 L if you make it 9 $\mathrm{cm}$ deep and give the sphere a radius of
16 $\mathrm{cm} .$ To be sure, you picture the wok as a solid of revolution, as
shown here, and calculate its volume with an integral. To the
nearest cubic centimeter, what volume do you really get?
$\left(1 \mathrm{L}=1000 \mathrm{cm}^{3}\right)$

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Problem 58

Max-min The arch $y=\sin x, 0 \leq x \leq \pi,$ is revolved about
the line $y=c, 0 \leq c \leq 1,$ to generate the solid in the accompanying figure.
\begin{equation}
\begin{array}{l}{\text { a. Find the value of } c \text { that minimizes the volume of the solid. }} \\ {\text { What is the minimum volume? }} \\ {\text { b. What value of } c \text { in }[0,1] \text { maximizes the volume of the solid? }}\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}{\text { c. Graph the solid's volume as a function of } c, \text { first for }} \\ {0 \leq c \leq 1 \text { and then on a larger domain. What happens to }} \\ {\text { the volume of the solid as } c \text { moves away from }[0,1] ? \text { Does }} \\ {\text { this make sense physically? Give reasons for your answers. }}\end{array}
\end{equation}

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Problem 59

Consider the region $R$ bounded by the graphs of $y=f(x)>0$
$x=a>0, x=b>a,$ and $y=0$ (see accompanying figure).
If the volume of the solid formed by revolving $R$ about the $x$ -axis
is $4 \pi,$ and the volume of the solid formed by revolving $R$ about
the line $y=-1$ is $8 \pi,$ find the area of $R .$

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Problem 60

Consider the region $R$ given in Exercise $63 .$ If the volume of the
solid formed by revolving $R$ around the $x$ -axis is $6 \pi,$ and the vol-
ume of the solid formed by revolving $R$ around the line $y=-2$ is
$10 \pi,$ find the area of $R .$

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