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Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 6

Applications of Definite Integrals

Educators


Problem 1

Find the volumes of the solids in Exercises $1-10$
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

Find the volumes of the solids in Exercises $1-10$
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 semi- circle $y=-\sqrt{1-x^{2}}$ 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

Find the volumes of the solids in Exercises $1-10$ .
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.

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Problem 6

Find the volumes of the solids in Exercises $1-10$ .
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
a. circular disks with diameters running from the curve
$\quad y=\tan x$ to the curve $y=\sec x$ .
b. squares whose bases run from the curve $y=\tan x$ to the curve $y=\sec x .$

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Problem 7

Find the volumes of the solids in Exercises $1-10$ .
The base of a solid is the region bounded by the graphs of $y=3 x, y=6,$ and $x=0 .$ The cross-scctions perpendicular to the $x$ -axis are
a. rectangles of height 10 .
b. rectangles of perimeter 20 .

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Problem 8

Find the volumes of the solids in Exercises $1-10$ .
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
a. isosceles triangles of height 6.
b. semi-circles with diameters running across the base of the solid.

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Problem 9

Find the volumes of the solids in Exercises $1-10$ .
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 circular disks with diameters running from the $y$ -axis to the parabola $x=\sqrt{5} y^{2}$

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Problem 10

Find the volumes of the solids in Exercises $1-10$ .
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 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.
a. Find the volume of the column.
b. What will the volume be if the square turns twice instead of once? Give reasons for your answer.

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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.
About the $x$ -axis

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Problem 16

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

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Problem 17

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

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Problem 18

In Exercises 15-18 find the volume of the solid generated by revolving the shaded region about the given axis.
About the $x$ -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 $19-24$ about the $x$ -axis.

$y=x^{2}, \quad y=0, \quad x=2$

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Problem 20

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $19-24$ about the $x$ -axis.
$y=x^{3}, \quad y=0, \quad x=2$

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Problem 21

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $19-24$ about the $x$ -axis.
$y=\sqrt{9-x^{2}}, \quad y=0$

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Problem 22

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $19-24$ about the $x$ -axis.
$y=x-x^{2}, \quad y=0$

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Problem 23

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $19-24$ about the $x$ -axis.
$y=\sqrt{\cos x}, \quad 0 \leq x \leq \pi / 2, \quad y=0, \quad x=0$

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Problem 24

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $19-24$ about the $x$ -axis.
$y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4$

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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.
The region in the first quadrant bounded above by the line $y=\sqrt{2},$ below by the curve $y=\sec x \tan x,$ and on the left by the $y$ -axis, about the line $y=\sqrt{2}$

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Problem 26

In Exercises 25 and $26,$ find the volume of the solid generated by revolving the region about the given line.
The region in the first quadrant bounded above by the line $y=2$ , below by the curve $y=2 \sin x, 0 \leq x \leq \pi / 2,$ and on the left by the $y$ -axis, about the line $y=2$

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Problem 27

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.

The region enclosed by $x=\sqrt{5} y^{2}, \quad x=0, \quad y=-1, \quad y=1$

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Problem 28

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.
The region enclosed by x=y^{3 / 2}, \quad x=0, \quad y=2

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Problem 29

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.
The region enclosed by $x=\sqrt{2 \sin 2 y}$ $0 \leq y \leq \pi / 2, \quad x=0$

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Problem 30

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.
The region enclosed by x=\sqrt{\cos (\pi y / 4)}, $-2 \leq y \leq 0$
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.
$x=2 /(y+1), \quad x=0, \quad y=0, \quad y=3$

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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.
$x=\sqrt{2 y} /\left(y^{2}+1\right), \quad x=0, \quad y=1$

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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 in Exercises 33 and 34 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 3540 about the $x$ -axis.
$y=x, \quad y=1, \quad x=0$

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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 3540 about the $x$ -axis.
$y=2 \sqrt{x}, \quad y=2, \quad x=0$

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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 3540 about the $x$ -axis.
$y=x^{2}+1, \quad y=x+3$

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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 3540 about the $x$ -axis.
$y=4-x^{2}, \quad y=2-x$

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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 3540 about the $x$ -axis.
$y=\sec x, \quad y=\sqrt{2}, \quad-\pi / 4 \leq x \leq \pi / 4$

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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 3540 about the $x$ -axis.
$y=\sec x, \quad y=\tan x, \quad x=0, \quad x=1$

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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.
$y=x, \quad y=1, \quad x=0$

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Problem 42

In Exercises $41-44$ , find the volume of the solid generated by revolving each region about the $y$ -axis.
The region cnclosed by the triangle with vertices $(0,1),(1,0),$ and $(1,1)$

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Problem 43

In Exercises $41-44$ , find the volume of the solid generated by revolving each region about the $y$ -axis.
The region in the first quadrant bounded above by the parabola $y=x^{2},$ below by the $x$ -axis, and on the right by the line $x=2$

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Problem 44

In Exercises $41-44$ , find the volume of the solid generated by revolving each region about the $y$ -axis.
The region in the first quadrant bounded on the left by the circle $x^{2}+y^{2}=3,$ on the right by the line $x=\sqrt{3},$ and above by the ine $y=\sqrt{3}$

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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.
The region in the first quadrant bounded above by the curve
$y=x^{2}$ , below by the $x$ -axis, and on the right by the line $x=1$
about the line $x=-1$

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Problem 46

In Exercises 45 and $46,$ find the volume of the solid generated by revolving each region about the given axis.
The region in the scond quadrant bounded above by the curve $y=-x^{3},$ below by the $x$ -axis, and on the left by the line $x=-1$ about the line $x=-2$

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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
a. the $x$ -axis.
c. the line $y=2, \quad$ d. the line $x=4$ .

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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
a. the line $x=1 . \quad$ b. the line $x=2$

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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
a. the line $y=1 . \quad$ b. the line $y=2$ .
c. the line $y=-1 .$

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Problem 50

By integration, find the volume of the solid generated by re-
volving the triangular region with vertices $(0,0),(b, 0),(0, h)$
about
a. the $x$ -axis. $\quad$ b. the $y$ -axis.

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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 $y=x^{2} / 2$ betwecn $y=0$ and $y=5$ about
the $y$ -axis.
a. Find the volume of the bowl.
b. Related rates If we fill the bowl with water at a constant
rate of 3 cubic units per second, how fast will the water level
in the bowl be rising when the water is 4 units decp?

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Problem 53

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

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Problem 54

Explain how you could estimate the volume of a solid of revolu-
tion 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 $\mathrm{g}$ , 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 $\mathrm{cm}$ decp 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.
a. Find the value of $c$ that minimizes the volume of the solid. What is the minimum volume?
b. What value of $c$ in $[0,1]$ maximizes the volume of the solid?
c. Graph the solid's volume as a function of $c$ , first for $0 \leq c \leq 1$ and then on a larger domain. What happens to the volume of the solid as $c$ moves away from $[0,1] ?$ Does this make sense physically? Give reasons for your answers.

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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 accomanying 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 $59 .$ If the volume of the solid formed by revolving $R$ around the $x$ -axis is $6 \pi,$ and the volume of the solid formed by revolving $R$ around the line $y=-2$ is $10 \pi,$ find the area of $R .$

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