Explanation Obtain the radius and height of the shells by rotation to find the volume by Shell Method. Given: Let R be the region under the graph of y = 9 − x 2 for 0 ≤ x ≤ 2 . Calculation: The region R is sketched below When this region is rotated about the x -axis, we obtain a solid with two different shells. For the area between [ 0 , 5 ] , the radius of the shell is y and height 2 and for the area between [ 5 , 9 ] , the radius of the shell will be y and height 9 − y . The volume of the solid using Shell Method is thus, V = 2 π ∫ 0 5 2 y d y + 2 π ∫ 5 9 y ( 9 − y ) d

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.4, Problem 33E

To determine

The volume of rotation of $R$ about the $x$ -axis using the Shell Method, as a sum of two integrals along the $y$ -axis using the hint that the shells generated depend on whether $y$ belongs to $[0, 5]$ or $[5, 9]$

Expert Solution & Answer

Chapter 6 Solutions

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Ch. 6.1 - Prob. 1PQ Ch. 6.1 - Prob. 2PQ Ch. 6.1 - Prob. 3PQ Ch. 6.1 - Prob. 4PQ Ch. 6.1 - Prob. 5PQ Ch. 6.1 - Prob. 6PQ Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - 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The volume of rotation of R about the x -axis using the Shell Method, as a sum of two integrals along the y -axis using the hint that the shells generated depend on whether y belongs to [ 0 , 5 ] or [ 5 , 9 ]

Solution Summary: The author explains how to determine the volume of rotation of R about the x -axis using the Shell Method.

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The volume of rotation of $R$ about the $x$ -axis using the Shell Method, as a sum of two integrals along the $y$ -axis using the hint that the shells generated depend on whether $y$ belongs to $[0, 5]$ or $[5, 9]$

Chapter 6 Solutions

The volume of rotation of R about the x -axis using the Shell Method, as a sum of two integrals along the y -axis using the hint that the shells generated depend on whether y belongs to [ 0 , 5 ] or [ 5 , 9 ]

Solution Summary: The author explains how to determine the volume of rotation of R about the x -axis using the Shell Method.

Concept explainers

Videos

The volume of rotation of R about the x-axis using the Shell Method, as a sum of two integrals along the y-axis using the hint that the shells generated depend on whether y belongs to [0,5] or [5,9]

Chapter 6 Solutions

The volume of rotation of $R$ about the $x$ -axis using the Shell Method, as a sum of two integrals along the $y$ -axis using the hint that the shells generated depend on whether $y$ belongs to $[0, 5]$ or $[5, 9]$