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  • Calculus: Early Transcendentals
  • Applications of Integration

Calculus: Early Transcendentals

James Stewart

Chapter 6

Applications of Integration - all with Video Answers

Educators

+ 19 more educators

Section 2

Volumes

03:35

Problem 1

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x + 1 $ , $ y = 0 $ , $ x = 0 $ , $ x = 2 $ ; about the x-axis

Carson Merrill
Carson Merrill
Numerade Educator
02:33

Problem 2

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \frac{1}{x} $ , $ y = 0 $ , $ x = 1 $ , $ x = 4 $ ; about the x-axis

Madi Sousa
Madi Sousa
Numerade Educator
03:30

Problem 3

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \sqrt{x - 1} $ , $ y = 0 $ , $ x = 5 $ ; about the x-axis

Chris Trentman
Chris Trentman
Numerade Educator
03:43

Problem 4

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = e^x $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $ ; about the x-axis

Chris Trentman
Chris Trentman
Numerade Educator
03:40

Problem 5

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = 2 \sqrt{y} $ , $ x = 0 $ , $ y = 9 $ ; about the y-axis

RG
Raymond Guo
Numerade Educator
06:06

Problem 6

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ 2x = y^2 $ , $ x = 0 $ , $ y = 4 $ ; about the y-axis

Chris Trentman
Chris Trentman
Numerade Educator
05:38

Problem 7

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = x $ , $ x \ge 0 $ ; about the x-axis

RG
Raymond Guo
Numerade Educator
04:28

Problem 8

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = 6 - x^2 $ , $ y = 2 $ ; about the x-axis

Chris Trentman
Chris Trentman
Numerade Educator
05:30

Problem 9

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y^2 = x $ , $ x = 2y $ ; about the y-axis

RG
Raymond Guo
Numerade Educator
06:40

Problem 10

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = -2 - y^2 $ , $ x = y^4 $ ; about the y-axis

Chris Trentman
Chris Trentman
Numerade Educator
04:16

Problem 11

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^2 $ , $ x = y^2 $ ; about $ y = 1 $

Chris Trentman
Chris Trentman
Numerade Educator
08:07

Problem 12

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = 1 $ , $ x = 2 $ ; about $ y = -3 $

Chris Trentman
Chris Trentman
Numerade Educator
06:54

Problem 13

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = 1 + \sec x $ , $ y = 3 $ ; about $ y = 1 $

Chris Trentman
Chris Trentman
Numerade Educator
08:09

Problem 14

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \sin x $ , $ y = \cos x $ , $ 0 \le x \le \frac{\pi}{4} $ ; about $ y = -1 $

Chris Trentman
Chris Trentman
Numerade Educator
05:36

Problem 15

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = 0 $ , $ x = 1 $ ; about $ x = 2 $

Chris Trentman
Chris Trentman
Numerade Educator
08:25

Problem 16

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ xy = 1 $ , $ y = 0 $ , $ x = 1 $ , $ x = 2 $ ; about $ x = -1 $

Chris Trentman
Chris Trentman
Numerade Educator
07:13

Problem 17

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ x = y^2 $ , $ x = 1 - y^2 $ ; about $ x = 3 $

Chris Trentman
Chris Trentman
Numerade Educator
03:58

Problem 18

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$$
y=x, y=0, x=2, x=4 ; \quad \text { about } x=1
$$

Mary Wakumoto
Mary Wakumoto
Numerade Educator
01:50

Problem 19

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_1 $ about $ OA $

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
02:51

Problem 20

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_1 $ about $ OC $

Chris Trentman
Chris Trentman
Numerade Educator
01:54

Problem 21

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_1 $ about $ AB $

Chris Trentman
Chris Trentman
Numerade Educator
04:02

Problem 22

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_1 $ about $ BC $

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
08:54

Problem 23

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_2 $ about $ OA $

Aparna Shakti
Aparna Shakti
Numerade Educator
02:36

Problem 24

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_2 $ about $ OC $

Chris Trentman
Chris Trentman
Numerade Educator
02:47

Problem 25

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_2 $ about $ AB $

Chris Trentman
Chris Trentman
Numerade Educator
01:54

Problem 26

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_2 $ about $ BC $

Chris Trentman
Chris Trentman
Numerade Educator
02:49

Problem 27

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_3 $ about $ OA $

Chris Trentman
Chris Trentman
Numerade Educator
02:09

Problem 28

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_3 $ about $ OC $

Chris Trentman
Chris Trentman
Numerade Educator
02:16

Problem 29

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_3 $ about $ AB $

Chris Trentman
Chris Trentman
Numerade Educator
02:16

Problem 30

Refer to the figure and find the volume generated by rotating the given region about the specified line.

$ \Re_3 $ about $ BC $

Chris Trentman
Chris Trentman
Numerade Educator
View

Problem 31

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = e^{-x^2} $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $

(a) About the x-axis
(b) About $ y = -1 $

SO
Sari Ogami
Numerade Educator
03:00

Problem 32

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = 0 $ , $ y = \cos^2 x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $

(a) About the x-axis
(b) About $ y = 1 $

Carson Merrill
Carson Merrill
Numerade Educator
10:44

Problem 33

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ x^2 + 4y^2 = 4 $

(b) About $ y = 2 $
(b) About $ x = 2 $

Linda Hand
Linda Hand
Numerade Educator
08:56

Problem 34

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y =x^2 $ , $ x^2 + y^2 = 1 $ , $ y \ge 0 $

(a) About the x-axis
(b) About the y-axis

Chris Trentman
Chris Trentman
Numerade Educator
06:09

Problem 35

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves.

$ y = \ln (x^6 + 2) $ , $ y = \sqrt{3 - x^3} $

Chris Trentman
Chris Trentman
Numerade Educator
03:25

Problem 36

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the x-axis the region bounded by these curves.

$ y = 1 + xe^{-x^3} $ , $ y = \arctan x^2 $

Chris Trentman
Chris Trentman
Numerade Educator
05:46

Problem 37

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$ y = \sin^2 x $ , $ y = 0 $ , $ 0 \le x \le \pi $ ; about $ y = -1 $

Aparna Shakti
Aparna Shakti
Numerade Educator
03:47

Problem 38

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$ y = x $ , $ y = xe^{1 - \frac{x}{2}} $ ; about $ y = 3 $

Chris Trentman
Chris Trentman
Numerade Educator
05:30

Problem 39

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{0}^\pi \sin x dx $

Michael Cooper
Michael Cooper
Numerade Educator
04:44

Problem 40

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{-1}^1 (1 -y^2)^2 dy $

Yuki Hotta
Yuki Hotta
Numerade Educator
03:01

Problem 41

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{0}^1 (y^4 - y^8) dy $

Linda Hand
Linda Hand
Numerade Educator
00:28

Problem 42

Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{1}^4 [3^2 - (3 - \sqrt{x})^2] dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:10

Problem 43

A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.

AL
Andrew Lebedinsky
Numerade Educator
00:29

Problem 44

A log 10 m long is cut at 1-meter intervals and its cross- sectional areas $ A $ (at a distance x from the end of the log) are listed in the table. Use the Midpoint Rule with $ n = 5 $ to estimate the volume of the log.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
07:50

Problem 45

(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with $ n = 4 $ to estimate the volume of the solid.
(b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with $ n = 4 $.

Linda Hand
Linda Hand
Numerade Educator
10:28

Problem 46

(a) A model for the shape of the bird's egg is obtained by rotating about the x-axis the region under the graph of
$$ f(x) = (ax^3 + bx^2 + cx + d) \sqrt{1 - x^2} $$
Use $ CAS $ to find the volume of such an egg.
(b) For a red-throated loon, $ a = -0.06 $, $ b = 0.04 $, $ c = 0.1 $, and $ d = 0.54 $. Graph $ f $ and find the volume of an egg of this species.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
View

Problem 47

Find the volume of the described solid $ S $.
A right circular cone with height $ h $ and base radius $ r $.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
05:56

Problem 48

Find the volume of the described solid $ S $.
A frustum of a right circular cone with height $ h $, lower base radius $ R $, and top radius $ r $.

Chris Trentman
Chris Trentman
Numerade Educator
06:31

Problem 49

Find the volume of the described solid $ S $.
A cap of a sphere with radius $ r $ and height $ h $.

Chris Trentman
Chris Trentman
Numerade Educator
08:08

Problem 50

Find the volume of the described solid $ S $.
A frustum of a pyramid with square base of side $ b $, square top of side $ a $, and height $ h $

What happens if $ a = b $? What happens if $ a = 0 $?

Cindy Rodgers
Cindy Rodgers
Numerade Educator
02:03

Problem 51

Find the volume of the described solid $ S $.
A pyramid with height $ h $ and rectangular base with dimensions $ b $ and $ 2b $.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:29

Problem 52

Find the volume of the described solid $ S $.
A pyramid with height $ h $ and base an equilateral triangle with side $ a $ (a tetrahedron).

Thomas Waite
Thomas Waite
Numerade Educator
07:29

Problem 53

Find the volume of the described solid $ S $.
A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm.

Chris Trentman
Chris Trentman
Numerade Educator
03:55

Problem 54

Find the volume of the described solid $ S $.
The base of $ S $ is a circular disk with radius $ r $. Parallel cross sections perpendicular to the base are squares.

Chris Trentman
Chris Trentman
Numerade Educator
04:17

Problem 55

Find the volume of the described solid $ S $.
The base of $ S $ is an elliptical region with boundary curve $ 9x^2 + 4y^2 = 36 $. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

Chris Trentman
Chris Trentman
Numerade Educator
05:20

Problem 56

Find the volume of the described solid $ S $.
The base of $ S $ is the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (0, 1) $. Cross-sections perpendicular to the y-axis are equilateral triangles.

Chris Trentman
Chris Trentman
Numerade Educator
02:42

Problem 57

Find the volume of the described solid $ S $.
The base of $ S $ is the same base as in Exercise 56, but cross-sections perpendicular to the x-axis are squares.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
02:30

Problem 58

Find the volume of the described solid $ S $.
The base of $ S $ is the region enclosed by the parabola $ y = 1 - x^2 $ and the x-axis. Cross-sections perpendicular to the y-axis are squares.

Chris Trentman
Chris Trentman
Numerade Educator
02:44

Problem 59

Find the volume of the described solid $ S $.
The base of $ S $ is the same base as in Exercise 58, but cross-sections perpendicular to the x-axis are isosceles triangles with height equal to the base.

Chris Trentman
Chris Trentman
Numerade Educator
02:51

Problem 60

Find the volume of the described solid $ S $.
The base of $ S $ is the region enclosed by $ y = 2 - x^2 $ and the x-axis. Cross-sections perpendicular to the y-axis are quarter-circles.

Chris Trentman
Chris Trentman
Numerade Educator
14:18

Problem 61

Find the volume of the described solid $ S $.
The solid $ S $ is bounded by circles that are perpendicular to the x-axis, intersect the x-axis, and have centers on the parabola $ y = \frac{1}{2} (1 - x^2) $, $ -1 \le x \le 1 $.

Michael Cooper
Michael Cooper
Numerade Educator
03:03

Problem 62

The base of $ S $ is a circular disk with radius $ r $. Parallel cross-sections perpendicular to the base are isosceles triangles with height $ h $ and unequal side in the base.
(a) Set up an integral for the volume of $ S $.
(b) By interpreting the integral as an area, find the volume of $ S $.

Carson Merrill
Carson Merrill
Numerade Educator
02:52

Problem 63

(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii $ r $ and $ R $.
(b) By interpreting the integral as an area, find the volume of the torus.

Carson Merrill
Carson Merrill
Numerade Educator
06:14

Problem 64

Solve Example 9 taking cross-sections to be parallel to the line of intersection of the two planes.

Chris Trentman
Chris Trentman
Numerade Educator
03:00

Problem 65

(a) Cavalieri's Principle states that if a family of parallel planes gives equal cross-section areas for two solids $ S_1 $ and $ S_2 $ then the volumes of $ S_1 $ and $ S_2 $ are equal. Prove this principle.
(b) Use Cavalieri's Principle to find the volume of the oblique cylinder shown in the figure.

Carson Merrill
Carson Merrill
Numerade Educator
01:18

Problem 66

Find the volume common to two circular cylinders, each with radius $ r $, if the axes of the cylinders intersect at right angles.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
09:05

Problem 67

Find the volume common to two spheres, each with radius $ r $, if the center of each sphere lies on the surface of the other sphere.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
15:02

Problem 68

A bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of $ h $ centimeters. Find the volume of water in the bowl.

DM
David Mccaslin
Numerade Educator
03:10

Problem 69

A hole of radius $ r $ is bored through the middle of a cylinder of radius $ R > r $ at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.

Aparna Shakti
Aparna Shakti
Numerade Educator
03:19

Problem 70

A hole of radius $ r $ is bored through the center of a sphere of radius $ R > r $. Find the volume of the remaining portion of the sphere.

Carson Merrill
Carson Merrill
Numerade Educator
10:34

Problem 71

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the
volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.
(a) A barrel with height $ h $ and maximum radius $ R $ is constructed by rotating about the x-axis the parabola $ y = R - cx^2 $, $ \frac{-h}{2} \le x \le \frac{h}{2} $, where c is a positive constant. Show that the radius of each end of the barrel is $ r = R - d $, where $ d = \frac{ch^2}{4} $.
(b) Show that the volume enclosed by the barrel is
$$ V = \frac{1}{3} \pi h (2R^2 + r^2 - \frac{2}{5} d^2) $$

Linda Hand
Linda Hand
Numerade Educator
05:14

Problem 72

Suppose that a region $ \Re $ has area $ A $ and lies above the x-axis. When $ \Re $ is rotated about the x-axis, it sweeps out a solid with volume $ V_1 $. When $ \Re $ is rotated about the line $ y = -k $ (where $ k $ is a positive number), it sweeps out a solid with volume $ V_2 $. Express $ V_2 $ in terms of $ V_1 $, $ k $, and $ A $.

Chris Trentman
Chris Trentman
Numerade Educator

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