L38 Volumes Shell - y = x 2 x and y = 0 about the line x = 3 7 6 6 NYTI 1 Determine the volume of the solid obtained by rotating the region bounded by y

L38 Volumes Shell - y = x 2 x and y = 0 about the line x =...

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L38 Volume of Solid of Revolution II{ Shell Method Shell Method is another way to calculate the volume of a solid of revolution when the slice is parallel to the axis of revolution and you integrate perpendicular to the axis of revolution. V shell = 2 rhdx = 2 (shell-radius)(shell-height) dx 1
The necessary equation for calculating such a volume depends on which axis is serving as the axis of revolution. V ver = Z A ( x ) dx if the rotation is around a vertical axis of revolution. V hor = Z A ( y ) dy if the rotation is around a horizontal axis of revolu- tion; Note: A shell = 2 (shell-radius)(shell-height) 2
ex. Determine the volume of the solid obtained by rotating the region bounded by y = ( x 1)( x 3) 2 and the x axis about the y -axis. ( 24 5 ) 3
ex. Determine the volume of the solid obtained by rotating the region bounded by y = x 2 + x and y = 0 about the line y axis. ( 6 ) 4
The remaining examples will have axis of rotation about axis other than the x , and y axis: ex. Determine the volume of the solid obtained by rotating the region bounded by x = ( y 2) 2 and y = x about the line y = 1. ( 63 2 ) 5
ex. Determine the volume of the solid obtained by rotating the region bounded by y = x 2 + x and y = 0 about the line x = 3. ( 7 6 ) 6
NYTI: 1. Determine the volume of the solid obtained by rotating the region bounded by y = p x and y = x about the line x = 4. ( 22 15 ) 2. Determine the volume of the solid obtained by rotating the region bounded by y = p x and y = x about the line x = 10. ( 16 5 ) 3. Determine the volume of the solid obtained by rotating the region bounded by y = p x and y = x about the line y = 3. ( 7 6 ) 7
4. Compute the volume of the solid obtained by ro- tating about the y axis the region bounded by the lines x = 0 ; x = 3 ; and y = 0 and the function f ( x ) = 3 x 3 x 4 . Using an alternative method, cylindrical shell method, (to nd the volume of the solid generated from re- volving a base area around some axis of rotation) sometimes is necessary especially when it is di cult to evaluate the base area with disk or washer meth- ods. Tonight’s homework: as usual, work out all NYTI problems. 8

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