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 $y=\sqrt{x}$ .

Check back soon!

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

Check back soon!

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

Check back soon!

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

Check back soon!

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.

Check back soon!

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 $y=\sec x .$

Check back soon!

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 .

Check back soon!

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.

Check back soon!

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 $x=\sqrt{5} y^{2}$

Check back soon!

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.

Check back soon!

Problem 11

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

Check back soon!

Problem 12

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

Check back soon!

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.

Check back soon!

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 $12 .$

Check back soon!

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

Check back soon!

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

Check back soon!

Problem 17

About the $y$ -axis

Check back soon!

Problem 18

About the $x$ -axis

Check back soon!

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.

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

Check back soon!

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.

$y=x^{3}, \quad y=0, \quad x=2$

Check back soon!

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.

$y=\sqrt{9-x^{2}}, \quad y=0$

Check back soon!

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.

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

Check back soon!

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.

$y=\sqrt{\cos x}, \quad 0 \leq x \leq \pi / 2, \quad y=0, \quad x=0$

Check back soon!

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.

$y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4$

Check back soon!

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 $y=\sqrt{2}$

Check back soon!

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 $y=2$

Check back soon!

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

Check back soon!

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

Check back soon!

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 $x=\sqrt{2 \sin 2 y}$ $0 \leq y \leq \pi / 2, \quad x=0$

Check back soon!

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

Check back soon!

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.

$x=2 /(y+1), \quad x=0, \quad y=0, \quad y=3$

Check back soon!

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.

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

Check back soon!

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

Check back soon!

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

Check back soon!

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.

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

Check back soon!

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.

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

Check back soon!

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.

$y=x^{2}+1, \quad y=x+3$

Check back soon!

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.

$y=4-x^{2}, \quad y=2-x$

Check back soon!

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.

$y=\sec x, \quad y=\sqrt{2}, \quad-\pi / 4 \leq x \leq \pi / 4$

Check back soon!

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.

$y=\sec x, \quad y=\tan x, \quad x=0, \quad x=1$

Check back soon!

Problem 41

In Exercises $41-44$ , find the volume of the solid generated by revolving each region about the $y$ -axis.

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

Check back soon!

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 $(1,1)$

Check back soon!

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 $x=2$

Check back soon!

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 $y=\sqrt{3}$

Check back soon!

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 $x=-1$

Check back soon!

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 $x=-2$

Check back soon!

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 $x=4$ .

Check back soon!

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 $x=2$

Check back soon!

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 $y=-1 .$

Check back soon!

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.

Check back soon!

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.)

Check back soon!

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?

Check back soon!

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?

Check back soon!

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.

Check back soon!

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.

Check back soon!

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)?

Check back soon!

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

Check back soon!

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.

Check back soon!

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 $R .$

Check back soon!

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 $R .$

Check back soon!