1. C YLINDRICAL SHELLS C ONTINUED 4-Jcont Comparison of Methods

2. Comparison of Disk and Shell Methods The disk and shell methods can be distinguished as follows. For the disk method, the representative rectangle is always perpendicular to the axis of revolution, whereas for the shell method, the representative rectangle is always parallel to the axis of revolution.

3. Finding volumes of Solids revolved about a horizontal axis or about a vertical axis Disk Washers Shells Disk Washers Shells

4. Comparison Disk & Washer method Disks and Washers (slices) to Axis of revolution Revolution about a horizontal axis are in terms of x Revolution about a vertical axis are in terms of y Cylindrical Shell method Shells (slices) are to the axis of revolution Revolutions about a horizontal axis are in terms of y Revolutions about a vertical axis are in terms of x.

5. 5. Find the volume of the solid formed by revolving the region bounded by the graphs of y = 3 + x 2 and y = 4 about the x- axis Which Method is Preferable y = 3 + x 2 y = 4

6. Which Method is Preferable 5. cont y = 3 + x 2 y = 4

7. Which Method is Preferable 6. Find the volume of the solid formed by revolving the region bounded by the graphs of y = x 2 + 1, y = 0, x = 0, and x = 1 about the y- axis. Solution: The washer method requires two integrals to determine the volume of this solid.

8. 6. cont’d

9. The shell method requires only one integral to find the volume.

10. Which Method is Preferable 7. Set up the volume of the solid formed by revolving the region bounded by the graphs of y = 1 + x 2 and y = x +3 about the x- axis. y = 1 + x 2 y = x + 3

11. Shell Method is Necessary 8. the volume of the solid formed by revolving the region bounded by the graphs of y = 1 + x + x 3 and y = 1 and x = 2 about the line x = 2 We can not easily solve y = 1 + x + x 3 for x in terms of y, therefore the variable of integration must be x. Choose a vertical reference rectangle. Because the rectangle is parallel to the axis of revolution, and use the shell method

12. H OME W ORK page 474 # 1-9, 15-19 (set up and use math 9), # 23 find volume, 21, 22, 25-26 no calculator, and 27-29