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Finding Volumes Disk/Washer/Shell

Finding Volumes Disk/Washer/Shell. Chapter 6.2 & 6.3 February 27, 2007. Review of Area: Measuring a length. Vertical Cut:. Horizontal Cut:. Disk Method. Slices are perpendicular to the axis of rotation. Radius is a function of position on that axis.

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Finding Volumes Disk/Washer/Shell

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  1. Finding VolumesDisk/Washer/Shell Chapter 6.2 & 6.3 February 27, 2007

  2. Review of Area: Measuring a length. Vertical Cut: Horizontal Cut:

  3. Disk Method • Slices are perpendicular to the axis of rotation. • Radius is a function of position on that axis. • Therefore rotating about x axis gives an integral in x; rotating about y gives an integral in y.

  4. Find the volume of the solid generated by revolving the region defined by , y = 8, and x = 0 about the y-axis. [0,8] Bounds? Length? Area? Volume?

  5. Find the volume of the solid generated by revolving the region defined by , and y = 1, about the line y = 1 [-1,1] Bounds? Length? Area? Volume?

  6. What if there is a “gap” between the axis of rotation and the function?

  7. Solids of Revolution: We determined that a cut perpendicular to the axis of rotation will either form a disk (region touches axis of rotation (AOR)) ora washer (there is a gap between the region and the AOR) Revolved around the line y = 1, the region forms a disk However when revolved around the x-axis, there is a “gap” between the region and the x-axis. (when we draw the radius, the radius intersects the region twice.)

  8. Area of a Washer Note: Both R and r are measured from the axis of rotation.

  9. Find the volume of the solid generated by revolving the region defined by , and y = 1, about the x-axis using planar slices perpendicular to the AOR. [-1,1] Bounds? Outside Radius? Inside Radius? Area? Volume?

  10. Find the volume of the solid generated by revolving the region defined by , and y = 1, about the line y=-1. [-1,1] Bounds? Outside Radius? Inside Radius? Area? Volume?

  11. Let R be the region in the x-y plane bounded by Set up the integral for the volume obtained by rotating R about the x-axis using planar slices perpendicular to the axis of rotation.

  12. Notice the gap: Outside Radius ( R ): Inside Radius ( r ): Area: Volume:

  13. Let R be the region in the x-y plane bounded bySet 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.

  14. Notice the gap: Outside Radius ( R ): Inside Radius ( r ): Area: Volume:

  15. Find the volume of the solid generated by revolving the region defined by , x = 3 and the x-axis about the x-axis. [0,3] Bounds? Length? (radius) Area? Volume?

  16. Note in the disk/washer methods, the focus in on the radius (perpendicular to the axis of rotation) and the shape it forms. We can also look at a slice that is parallel to the axis of rotation.

  17. Note in the disk/washer methods, the focus in on the radius (perpendicular to the axis of rotation) and the shape it forms. We can also look at a slice that is parallel to the axis of rotation. Length of slice Area:

  18. Volume = Slice is PARALLEL to the AOR

  19. Using on the interval [0,2] revolving around the x-axis using planar slices PARALLEL to the AOR, we find the volume: Length of slice? Radius? Area? Volume?

  20. Back to example:Find volume of the solid generated by revolving the region about the y-axis using cylindrical slices Length of slice ( h ): Radius ( r ): Area: Volume:

  21. Find the volume of the solid generated by revolving the region:about the y-axis, using cylindrical slices. Length of slice ( h ): Inside Radius ( r ): Area: Volume:

  22. Try: • Set up an integral integrating with respect to y to find the volume of the solid of revolution obtained when the region bounded by the graphs of y = x2 and y = 0 and x = 2 is rotated around • a) the y-axis • b) the line y = 4

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