Lesson Notes 6.5

1. Lesson Notes 6.5 Disks, Washers and Shells* * BC Topic Greg Kelly, Hanford High School, Richland, Washington

2. Example #1 Suppose you start with this curve. Your boss at the ACME Rocket Company has assigned you to build a nose cone in this shape. So you put a piece of wood in a lathe and turn it to a shape to match the curve.

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

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

5. This application of the method of slicing is called the disk method. (We looked at this method last lesson). The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. A shape rotated about the y-axis would be: Since the shape is rotated about the x-axis, the formula is: V= Since we will be using the disk method to rotate shapes about other lines besides the x-axis, you will NOT be given this formula on quizzes or tests.

6. Example #2 The region between the curve , and the y-axis is revolved about the y-axis. Find the volume. y x The radius is the x value of the function, which is . We use a horizontal disk. The thickness is dy. volume of disk

7. The volume of the washer is: Example #3 The region bounded by and is revolved about the y-axis. What is the volume? Using a horizontal slice, The “disk” now has a hole in it, making it a “washer”. outer radius inner radius

8. The washer method formula is: 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. Like the disk method, this formula will not be given for quizzes or tests. I need you to understand the formula. 

9. r R Example #3 If the same region is rotated about the line x=2: The outer Radius is: The inner radius is:

11. Find the volume of the region bounded by , , and revolved about the y-axis. cylinder inner radius outer radius thickness of slice

12. There is another way we could approach this problem. Warning….BC Topic… cross section If we take a vertical slice and revolve it about the y-axis we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object with a single integral.

13. 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.

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

15. Example #4 Find the volume generated when this shape is revolved about the y axis. Since we can’t solve for x, we can’t use a horizontal slice.

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

18. 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. Assignment: Text pg 407: #15; 17; 19; 25; 27 as wel 2 AP free response questions in homework package. 