Warm Up

1. Warm Up

2. 8.3B – Volume of Solids • Goal: Use the disk and washer method to find volumes.

3. Take Notes on the video below • http://education-portal.com/academy/lesson/how-to-find-volumes-of-revolution-with-integration.html#lesson

4. Example 1 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.

5. Cont. The volume of each flat cylinder (disk) is: 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. r= the y value of the function

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

7. Disk Method This application of the method of slicing is called the disk method. 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 x-axis would be: A shape rotated about the y-axis would be:

8. Example 2 The region between the curve , and the y-axis is revolved about the y-axis. Find the volume.

9. Solution

10. Washer Method 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.

11. Example 3 The volume of the washer is: The region bounded by and is revolved about the y-axis. Find the volume. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”.

12. Solution outer radius inner radius