1. Finding Volumes Disk/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. Bounds? Length? Area? Volume? [0,8]

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

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)) or a 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. Bounds? Outside Radius? Inside Radius? Area? [-1,1] Volume?

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

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. Bounds? Length? (radius) Area? Volume? [0,3]

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. Area: Length of slice

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: Radius? Length of slice? 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 = x 2 and y = 0 and x = 2 is rotated around a)the y-axis b)the line y = 4