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

2. 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.

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

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

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

6. 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.)

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

8. 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?
[-1,1]
Outside Radius?
Inside Radius?
Area?
Volume?

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

10. 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.

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

12. 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.

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

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

15. 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.

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.
Length of slice
Area:

17. Volume =
Slice is PARALLEL to the AOR

18. 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?

19. 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:

20. 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:

21. 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