2. Volume: The Shell Method 7.3 Copyright © Cengage Learning. All rights reserved.

3. Find the volume of a solid of revolution using the shell method. Compare the uses of the disk method and the shell method. Objectives

4. Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006 7.3 The Shell Method Japanese Spider Crab Georgia Aquarium, Atlanta

5. Find the volume of the region bounded by,, and revolved about the y - axis. We can use the washer method if we split it into two parts: outer radius inner radius thickness of slice cylinder Japanese Spider Crab Georgia Aquarium, Atlanta

6. If we take a vertical sliceand revolve it about the y-axis we get a cylinder. cross section If we add all of the cylinders together, we can reconstruct the original object. Here is another way we could approach this problem:

7. cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

8. cross section If we add all the cylinders from the smallest to the largest: This is called the shell method because we use cylindrical shells.

9. 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.

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

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

12. 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. 

13. p(x): distance between the axis of revolution and the center of the rectangle h(x): height of the rectangle Notice that the representative rectangle is parallel to the axis of revolution

14. Using the Shell Method to Find Volume Use the shell method when the axis of revolution is perpendicular to the axis containing the natural interval of integration. Instead of summing volumes of thin slices, we sum volumes of thin cylindrical shells that grow outward from the axis of revolution. 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.

15. You Try: Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x 3 and the x- axis (0 ≤ x ≤ 1) about the y- axis. Solution: Because the axis of revolution is vertical, use a vertical representative rectangle, as shown in Figure 7.30. The width ∆x indicates that x is the variable of integration. Figure 7.30

16. Example 2 – Solution The distance from the center of the rectangle to the axis of revolution is p(x) = x, and the height of the rectangle is h(x) = x – x 3. Because x ranges from 0 to 1, the volume of the solid is cont’d

17. Comparison of Disk and Shell Methods Figure 7.32 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, as shown in Figure 7.32.

18. Implies everything is in terms of y. Diagram Shell Method

20. Diagram

22. Homework 7.3 pg.472 #1-27 odd