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MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace,

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1 MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.3 – Volumes by Cylindrical Shells Copyright © 2006 by Ron Wallace, all rights reserved.

2 Reminder: Definition of a Definite Integral Simplified Version where …

3 Cylindrical Shells h r t If t is small …

4 f(x) ba Volumes Using Cylindrical Shells xkxk radius of cylinder height of cylinder circumference of cylinder thickness of cylinder Cross-section is taken parallel to the line of rotation.

5 Solids of Revolution - Example A solid obtained by revolving a region around a line. Find the volume of a cone of height h and bottom radius r. r h NOTE: Cross-section is parallel to the axis of rotation.

6 Volumes of Solids of Revolution Comparing Methods Disks & WashersCylindrical Shells Cross-section: Perpendicular to the line of rotation. Cross-section: Parallel to the line of rotation. r is the length of the cross- section in terms of x or y. Rotate about the x-axis: Rotate about the y-axis: h is the length of the cross- section in terms of x or y. Rotate about the x-axis: Rotate about the y-axis:

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