RedAssign7[1].2 - space it passes through makes up the...

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Unit: Applications of Integration Module: Disks and Washers Solids of Revolution [email protected] Copyright 2001, Thinkwell Corp. All Rights Reserved. 1587 –rev 08/29/2001 Revolving a plane region about a line forms a solid of revolution . The volume of a solid of revolution using the disk method where R ( x ) is the radius of the solid of revolution with respect to x is V where: = π 2 [() ] b a VR x d x . Some solids can be described by moving regions through space as well as by slicing the solid into pieces. Consider the plane region given to the left. What happens if you rotate that region around the x -axis? To visualize the solid, think of the region as though it were connected to the x -axis on a hinge. As the region moves through space around the hinge, the
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Unformatted text preview: space it passes through makes up the solid. A solid defined in this way is called a solid of revolution . To find the volume of a solid of revolution, you can sometimes divide the region into slices. Each slice resembles a disk, so this method is called the disk method . To find the volume, just integrate the areas of the disks across the given interval. The radius of a given disk is equal to the height of the original region. The area of a disk equals the area of a circle. Once you find the area, just integrate. Notice that sometimes you can find shortcuts in the integral based upon symmetry or other properties of the region. Setting up the integral is the tough part of finding volumes. Once you have the integral, evaluating it is a piece of cake....
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This note was uploaded on 04/27/2010 for the course MATH 172 taught by Professor Hyon during the Spring '10 term at Community College of Denver.

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