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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 3

Volumes by Cylindrical Shells

01:56

Problem 1

Let $ S $ be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume $ V $ of $ S $. Sketch a typical approximating shell. What are its circumference and height? Use shells to find $ V $.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:35

Problem 2

Let $ S $ be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of $ S $. Do you think this method is preferable to slicing? Explain.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:18

Problem 3

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = \sqrt[3]{x} $ , $ y = 0 $ , $ x = 1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:59

Problem 4

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

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

Amrita Bhasin
Amrita Bhasin
Numerade Educator
10:27

Problem 5

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

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

Leon Druch
Leon Druch
Numerade Educator
01:18

Problem 6

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = 4x - x^2 $ , $ y = x $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:33

Problem 7

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

$ y = x^2 $ , $ y = 6x - 2x^2 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:51

Problem 8

Let $ V $ be the volume of the solid obtained by rotating about the y-axis the region bounded by $ y = \sqrt{x} $ and y = x^2 $. Find $ V $ both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:16

Problem 9

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ xy = 1 $ , $ x = 0 $ , $ y = 1 $ , $ y = 3 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:19

Problem 10

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

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

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:13

Problem 11

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ y = x^{\frac{3}{2}} $ , $ y = 8 $ , $ x = 0 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:55

Problem 12

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ x = -3y^2 + 12y - 9 $ , $ x = 0 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:19

Problem 13

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ x = 1 + (y - 2)^2 $ , $ x = 2 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:04

Problem 14

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

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

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:03

Problem 15

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = x^3 $ , $ y = 8 $ , $ x = 0 $ ; about $ x = 3 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:12

Problem 16

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = 4 - 2x $ , $ y = 0 $ , $ x = 0 $ ; about $ x = -1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:46

Problem 17

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = 4x - x^2 $ , $ y = 3 $ ; about $ x = 1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:51

Problem 18

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ y = \sqrt{x} $ , $ x = 2y $ ; about $ x = 5 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:05

Problem 19

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$ x = 2y^2 $ , $ y \ge 0 $ , $ x = 2 $ ; about $ y = 2 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:16

Problem 20

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

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

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:54

Problem 21

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = xe^{-x} $ , $ y = 0 $ , $ x = 2 $ ; about the y-axis

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:55

Problem 22

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = \tan x $ , $ y = 0 $ , $ x = \frac{\pi}{4} $ ; about $ x = \frac{\pi}{2} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:34

Problem 23

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = \cos^4 x $ , $ y = -\cos^4 x $ , $ \frac{-\pi}{2} \le x \le \frac{\pi}{2} $ ; about $ x = \pi $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:34

Problem 24

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

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

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:41

Problem 25

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ x = \sqrt{\sin y} $ , $ 0 \le y \le \pi $ , $ x = 0 $ ; about $ y = 4 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:20

Problem 26

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ x^2 - y^2 = 7 $ , $ x = 4 $ ; about $ y = 5 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:29

Problem 27

Use the Midpoint Rule with $ n = 5 $ to estimate the volume obtained by rotating about the y-axis the region under the curve $ y = \sqrt{1 + x^3} $ , $ 0 \le x \le 1 $.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:51

Problem 28

If the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with $ n = 5 $ to estimate the volume of the solid.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
00:39

Problem 29

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

$ \displaystyle \int_{0}^3 2 \pi x^5 dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:50

Problem 30

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

$ \displaystyle \int_{1}^3 2 \pi y \ln y dy $

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
00:42

Problem 31

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

$ 2 \pi \displaystyle \int_{1}^4 \frac{y + 2}{y^2} dy $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:36

Problem 32

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

$ \displaystyle \int_{0}^1 2 \pi (2 - x)(3^x - 2^x) dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:17

Problem 33

Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.

$ y = x^2 - 2x $ , $ y = \frac{x}{x^2 + 1} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:19

Problem 34

Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.

$ y = e^{\sin x} $ , $ y = x^2 - 4x + 5 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:38

Problem 35

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 = \sin^4 x $ , $ 0 \le x \le \pi $ ; about $ x = \frac{\pi}{2} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:55

Problem 36

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^3 \sin x $ , $ y = 0 $ , $ 0 \le x \le \pi $ ; about $ x = -1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:30

Problem 37

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ y = -x^2 + 6x - 8 $ , $ y = 0 $ ; about the y-axis

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:23

Problem 38

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ y = -x^2 + 6x - 8 $ , $ y = 0 $ ; about the x-axis

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:05

Problem 39

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

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

WZ
Wen Zheng
Numerade Educator
02:25

Problem 40

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

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

WZ
Wen Zheng
Numerade Educator
01:38

Problem 41

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x^2 + (y - 1)^2 = 1 $ ; about the y-axis

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:03

Problem 42

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x = (y - 3)^2 $ , $ x = 4 $ ; about $ y = 1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:41

Problem 43

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x = (y - 1)^2 $ , $ x - y = 1 $ ; about $ x = -1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:44

Problem 44

Let $ T $ be the triangular region with vertices $ (0, 0) $, $ (1, 0) $, and $ (1, 2) $, and let $ V $ be the volume of the solid generated when $ T $ is rotated about the line $ x = a $, where $ a > 1 $. Express $ a $ in terms of $ V $.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:19

Problem 45

Use cylindrical shells to find the volume of the solid.

$ A $ sphere of radius $ r $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:55

Problem 46

Use cylindrical shells to find the volume of the solid.

The solid torus of Exercise 6.2.63

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:15

Problem 47

Use cylindrical shells to find the volume of the solid.

$ A $ right circular cone with height $ h $ and base radius $ r $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:55

Problem 48

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height $ h $, as shown in the figure.
(a) Guess which ring has more wood in it.
(b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius $ r $ through the center of a sphere of radius $ R $ and express the answer in terms of $ h $.

Carson Merrill
Carson Merrill
Numerade Educator

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