1. Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 7.2 Disk and Washer Methods Limerick Nuclear Generating Station, Pottstown, Pennsylvania

2. Find the volume of a solid of revolution using the disk method. Find the volume of a solid of revolution using the washer method. Find the volume of a solid with known cross sections. Objectives

3. Integration as an Accumulation Process

4. The integration formula for the area between two curves was developed by using a rectangle as the representative element. Each integration formula will then be obtained by summing or accumulating these representative elements. For example, the area formula in this section was developed as follows. Integration as an Accumulation Process

5. Find the area of the region bounded by the graph of y = 4 – x 2 and the x-axis. Describe the integration as an accumulation process. Solution: The area of the region is given by You can think of the integration as an accumulation of the areas of the rectangles formed as the representative rectangle slides from x = –2 to x = 2, as shown in Figure 7.11. Describing Integration as an Accumulation Process

6. Solution Figure 7.11 cont’d

7. Volume: The Disk Method 2015 Copyright © Cengage Learning. All rights reserved. 7.2

8. The Disk Method

9. Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. So I put a piece of wood in a lathe and turn it to a shape to match the curve.

10. How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is: In this case: r= the y value of the function thickness = a small change in x = dx

11. The volume of each flat cylinder (disk) is: If we add the volumes, we get:

12. Solids of Revolution A solid of revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region; the line is called the axis of revolution. Many familiar solids are of this type.

13. Figure 7.15 The Disk Method

14. Example 1 – Using the Disk Method Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 ≤ x ≤ π) about the x-axis. Solution: From the representative rectangle in the upper graph in Figure 7.16, you can see that the radius of this solid is R(x) = f(x) Figure 7.16

15. Example 1 – Solution So, the volume of the solid of revolution is cont’d

16. The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

19. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. Draw the graph. Write x in terms of y. Volume of a cone?

20. 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.

21. AB/BC Homework: Pg. 463 #1-4all, 7-10 all, 11(a and c only),12(b and d only)