University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 6

Applications of Definite Integrals

Educators

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Problem 1

Find the volumes of the solids
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 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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Kevin M.
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Problem 3

Find the volumes of the solids 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 3

Find the volumes of the solids 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 $y=\sqrt{1-x^{2}}$

KM
Kevin M.
Numerade Educator

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. CANT COPY THE GRAPH

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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
a. circular disks with diameters running from the curve $y=\tan x$ to the curve $y=\sec x$
b. squares whose bases run from the curve $y=\tan x$ to the curve $y=\sec x$

KM
Kevin M.
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Problem 7

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

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Problem 8

Find the volume of solids
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. semicircles with diameters running across the base of the solid.

KM
Kevin M.
Numerade Educator

Problem 9

Find the volume of solids
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 volume of solids
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. CANT COPY THE GRAPH

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Kevin M.
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Problem 11

Find the volume of solids
Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.) CANT COPY THE GRAPH

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Problem 12

Find the volume of solids Find the volume of the given pyramid, which has a square base of area 9 and height 5 CANT COPY THE GRAPH

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Kevin M.
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Problem 13

A twisted solid $\quad$ 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 $\quad$ 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 CANT COPY THE GRAPH

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Kevin M.
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Problem 15

Intersection of two half-cylinders Two half-cylinders of diamcter 2 meet at a right angle in the accompanying figure. Find the volume of the solid region common to both half-cylinders. (Hint:
Consider slices parallel to the base of the solid.) CANT COPY THE GRAPH

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Problem 16

Gasoline in a tank $\quad$ A gasoline tank is in the shape of a right circular cylinder (lying on its side) of length $10 \mathrm{ft}$ and radius $4 \mathrm{ft}$. Set up an integral that represents the volume of the gas in the tank if it is filled to a depth of 6 ft. You will learn how to compute this integral in Chapter 8 (or you may use geometry to find its value). CANT COPY THE GRAPH

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Kevin M.
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Problem 17

Find the volume of the solid generated by revolving the shaded region about the given axis. About the $x$ -axis CANT COPY THE GRAPH

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Problem 18

Find the volume of the solid generated by revolving the shaded region about the given axis. About the $y$ -axis CANT COPY THE GRAPH

KM
Kevin M.
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Problem 19

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 20

Find the volume of the solid generated by revolving the shaded region about the given axis. About the $x$ -axis CANT COPY THE GRAPH

KM
Kevin M.
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Problem 21

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=x^{2}, \quad y=0, \quad x=2$$

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Problem 22

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=x^{3}, \quad y=0, \quad x=2$$

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Kevin M.
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Problem 23

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

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Problem 24

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

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Kevin M.
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Problem 25

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

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Problem 26

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4$$

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Kevin M.
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Problem 27

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=e^{-x}, \quad y=0, \quad x=0, \quad x=1$$

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Problem 28

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves The region between the curve $y=\sqrt{\cot x}$ and the $x$ -axis from $x=\pi / 6$ to $x=\pi / 2$

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Kevin M.
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Problem 29

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves The region between the curve $y=1 /(2 \sqrt{x})$ and the $x$ -axis from $x=1 / 4$ to $x=4$

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Problem 30

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves y=e^{x-1}, \quad y=0, \quad x=1, \quad x=3

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Kevin M.
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Problem 31

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 32

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 y-axis, abgut the line $y=2$

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Kevin M.
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Problem 33

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

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Problem 34

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

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Problem 35

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 36

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

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Kevin M.
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Problem 37

$x=2 / \sqrt{y+1}, \quad x=0, \quad y=0, \quad y=3$

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Problem 38

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

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Kevin M.
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Problem 39

Find the volumes of the solids generated by revolving the shaded regions
The $x$ -axis CANT COPY THE GRAPH

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Problem 40

Find the volumes of the solids generated by revolving the shaded regions
The $y$ -axis CANT COPY THE GRAPH

KM
Kevin M.
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Problem 41

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=x, \quad y=1, \quad x=0$$

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Problem 42

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=2 \sqrt{x}, \quad y=2, \quad x=0$$

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Kevin M.
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Problem 43

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=x^{2}+1, \quad y=x+3$$

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Problem 44

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=4-x^{2}, \quad y=2-x$$

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Kevin M.
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Problem 45

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves . $$y=\sec x, \quad y=\sqrt{2}, \quad-\pi / 4 \leq x \leq \pi / 4$$

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Problem 46

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves $$y=\sec x, \quad y=\tan x, \quad x=0, \quad x=1$$

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Kevin M.
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Problem 47

Find the volume of the solid generated by revolving each region about the $y$ -axis.
The region enclosed by the triangle with vertices (1,0),(2,1) and (1,1)

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Problem 48

Find the volume of the solid generated by revolving each region about the $y$ -axis.
The region enclosed by the triangle with vertices (0,1),(1,0) and (1,1)

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Kevin M.
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Problem 49

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 50

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 line $y=\sqrt{3}$

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Kevin M.
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Problem 51

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 52

Find the volume of the solid generated by revolving each region about the given axis.
The region in the second 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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Kevin M.
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Problem 53

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.
b. the $y$ -axis.
c. the line $y=2$
d. the line $x=4$

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Problem 54

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 $x=1$
b. the line $x=2$

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Kevin M.
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Problem 55

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 $y=1$
b. the line $y=2$
c. the line $y=-1$

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Problem 56

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

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Kevin M.
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Problem 57

The volume of a torus $\quad$ 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 58

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.
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 deep?

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Kevin M.
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Problem 59

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 m deep?

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Problem 60

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 light shining directly aboye it.

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Kevin M.
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Problem 61

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. CANT COPY THE GRAPH

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Problem 62

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)? CANT COPY THE GRAPH

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Kevin M.
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Problem 63

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 $3 \mathrm{L}$ 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? $\left(1 \mathrm{L}=1000 \mathrm{cm}^{3}\right)$ CANT COPY THE GRAPH

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Problem 64

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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Kevin M.
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Problem 65

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 $R$ CANT COPY THE GRAPH

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Problem 66

Consider the region $R$ given in Exercise $65 .$ 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 $R$

KM
Kevin M.
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