1. Review: Volumes of Revolution

2. x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

3. x y Even though we started with a cylinder,  does not enter the calculation!

4. Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections 

5. 7.3 Disk and Washer Methods

6. The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

7. The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius

8. This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formula is:

9. Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

10. Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

11. 7.3 The Shell Method

12. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

13. Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder.

14. Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

15. When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method. 

16. Volumes Using Cylindrical Shells The region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.

17. Volumes Using Cylindrical Shells The region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid using cylindrical shells.