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Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 6

Applications Of Definite Integrals

Educators


Problem 1

In Exercises 1 and $2,$ find a formula for the area $A(x)$ of the cross- sections of the solid perpendicular to the $x$ -axis.
The solid lies between planes perpendicular to the $x$ -axis at $x=-1$ and $x=1 .$ In each case, the cross-sections perpendicular to the $x$ -axis between these planes run from the semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$
a. The cross-sections are circular disks with diameters in the $x y$ -plane.
b. The cross-sections are squares with bases in the $x y$ -plane.
c. The cross-sections are squares with diagonals in the $x y$ -plane. (The length of a square's diagonal is $\sqrt{2}$ times the length of its sides.)
d. The cross-sections are equilateral triangles with bases in the $x y$ -plane.

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Problem 2

In Exercises 1 and $2,$ find a formula for the area $A(x)$ of the cross- sections of the solid perpendicular to the $x$ -axis.
The solid lies between planes perpendicular to the $x$ -axis at $x=0$ and $x=4 .$ The cross-sections perpendicular to the $x$ -axis between these planes run from the parabola $y=-\sqrt{x}$ to the parabola $y=\sqrt{x}$ .
a. The cross-sections are circular disks with diameters in the $x y$ -plane.
b. The cross-sections are squares with bases in the $x y$ -plane.
c. The cross-sections are squares with diagonals in the $x y$ -plane.
d. The cross-sections are equilateral triangles with bases in the $x y$ -plane.

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Problem 3

Find the volumes of the solids in Exercises $3-10$ .
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 4

Find the volumes of the solids in Exercises $3-10$ .
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 5

Find the volumes of the solids in Exercises $3-10$ .
The solid lies between planes perpendicular to the $x$ -axis at $x=-1$ and $x=1 .$ The cross-sections perpendicular to the $x$ -axis 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 6

Find the volumes of the solids in Exercises $3-10$ .
The solid lies between planes perpendicular to the $x$ -axis at $x=-1$ and $x=1 .$ The cross-sections perpendicular to the $x$ -axis 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 7

Find the volumes of the solids in Exercises $3-10$ .
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 figure.
b. squares with bases running from the $x$ -axis to the curve.

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Problem 8

Find the volumes of the solids in Exercises $3-10$ .
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 $y=\tan x$ 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 9

Find the volumes of the solids in Exercises $3-10$ .
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 $3-10$ .
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

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 12

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 13

In Exercises $13-16,$ 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 14

In Exercises $13-16,$ 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 15

In Exercises $13-16,$ 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 16

In Exercises $13-16,$ 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 17

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $17-22$ about the $x$ -axis.
$$
y=x^{2}, \quad y=0, \quad x=2
$$

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Problem 18

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $17-22$ about the $x$ -axis.
$$
y=x^{3}, \quad y=0, \quad x=2
$$

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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 $17-22$ about the $x$ -axis.
$$
y=\sqrt{9-x^{2}}, \quad y=0
$$

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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 $17-22$ about the $x$ -axis.
$$
y=x-x^{2}, \quad y=0
$$

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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 $17-22$ 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 22

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $17-22$ about the $x$ -axis.
$$
y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4
$$

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Problem 23

In Exercises 23 and $24,$ 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 24

In Exercises 23 and $24,$ 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 25

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $25-30$ 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 26

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $25-30$ about the $y$ -axis.
The region enclosed by $x=y^{3 / 2}, \quad x=0, \quad y=2$

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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 $25-30$ about the $y$ -axis.
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 28

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $25-30$ about the $y$ -axis.
The region enclosed by $x=\sqrt{\cos (\pi y / 4)}, \quad-2 \leq y \leq 0$ $x=0$

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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 $25-30$ about the $y$ -axis.
$$
x=2 /(y+1), \quad x=0, \quad y=0, \quad y=3
$$

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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 $25-30$ about the $y$ -axis.
$$
x=\sqrt{2 y} /\left(y^{2}+1\right), \quad x=0, \quad y=1
$$

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Problem 31

Find the volumes of the solids generated by revolving the shaded regions in Exercises 31 and 32 about the indicated axes.
The $x$ -axis

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Problem 32

Find the volumes of the solids generated by revolving the shaded regions in Exercises 31 and 32 about the indicated axes.
The $y$ -axis

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Problem 33

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $33-38$ about the $x$ -axis.
$$
y=x, \quad y=1, \quad x=0
$$

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Problem 34

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises $33-38$ about the $x$ -axis.
$$
y=2 \sqrt{x}, \quad y=2, \quad x=0
$$

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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 $33-38$ about the $x$ -axis.
$$
y=x^{2}+1, \quad y=x+3
$$

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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 $33-38$ about the $x$ -axis.
$$
y=4-x^{2}, \quad y=2-x
$$

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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 $33-38$ about the $x$ -axis.
$$
y=\sec x, \quad y=\sqrt{2}, \quad-\pi / 4 \leq x \leq \pi / 4
$$

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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 $33-38$ about the $x$ -axis.
$$
y=\sec x, \quad y=\tan x, \quad x=0, \quad x=1
$$

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Problem 39

In Exercises $39-42$ , 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 40

In Exercises $39-42$ , 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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Problem 41

In Exercises $39-42$ , 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 42

In Exercises $39-42$ , 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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Problem 43

In Exercises 43 and $44,$ 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 44

In Exercises 43 and $44,$ 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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Problem 45

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
$\begin{array}{ll}{\text { a. the } x \text { -axis. }} & {\text { b. the } y \text { -axis. }} \\ {\text { c. the line } y=2 .} & {\text { d. the line } x=4}\end{array}$

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Problem 46

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 47

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 48

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. $\quad$ b. the $y$ -axis.

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Problem 49

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 50

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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Problem 51

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 hemi- spherical 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 52

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 53

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 54

Volume of a cone Use calculus to find the volume of a right circular cone of height $h$ and base radius $r .$

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Problem 55

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? $\left(1 \mathrm{L}=1000 \mathrm{cm}^{3} .\right)$

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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 $190 \mathrm{g},$ 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

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 Figure 6.16 .
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 58

An auxiliary fuel tank You are designing an auxiliary fuel tank that will fit under a helicopter's fuselage to extend its range. After some experimentation at your drawing board, you decide to shape the tank like the surface generated by revolving the curve $y=1-\left(x^{2} / 16\right),-4 \leq x \leq 4,$ about the $x$ -axis (dimensions in feet).
a. How many cubic feet of fuel will the tank hold (to the nearest cubic foot)?
b. A cubic foot holds 7.481 gal. If the helicopter gets 2 $\mathrm{mi}$ to the gallon, how many additional miles will the helicopter be able to fly once the tank is installed (to the nearest mile)?

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