L11 - Volume Calculation by the Method of Rotation

L11 - Volume Calculation by the Method of Rotation -...

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Volume By Rotation (Section 6.1) The volume of a slice is the area of the slice multiplied by the width of the slice. V i = (area of side of slice i )(width i ). The total volume is then: V A ( x ) dx a b or V A ( y ) dy c d . We are going to examine slice perpendicular to an axis of revolution. When we revolve the slices, we get disks or washers depending on the initial region. Always make slices perpendicular to axis of revolution. Always integrate along the axis perpendicular to a slice. DEF The Disk Method T volume of a solid of revolution whose cross-sections are circular disks of radius R(x) is V A ( x ) dx a b [ R ( x )] 2 dx a b
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Ex. Find the volume of the solid generated by revolving the region bounded by y x 2 3 , the x-axis, and x = 8 around: (a) the x-axis (b) x = 8
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DEF The Washer Method The volume of a solid of revolution whose cross-sections are
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Unformatted text preview: circular washers of outer radius R(x) and inner radius r(x) is V A ( x ) dx a b [ R )] 2 r Ex. Find the volume of the solid generated by revolving the region bounded by y 3 , the x-axis, and x = 8 around: (a) the y-axis (b) y = -1 (c) y = 6 (d) x = -1 Do: 1. Let R be the region in the xy plane bounded by y 4 x , x 2, and y 1 4 . Set up the integral for the volume of the solid obtained by rotating R about the y-axis, using planar slices perpendicular to the axis of rotation. 2. Let R be the region in the xy plane bounded by y 2 x 2 and y 3 x 1 . Set up the integral for the volume of the solid obtained by rotating R about the x-axis, using planar slices perpendicular to the axis of rotation....
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This note was uploaded on 08/28/2011 for the course MATH 1206 taught by Professor Llhanks during the Spring '08 term at Virginia Tech.

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L11 - Volume Calculation by the Method of Rotation -...

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