1 / 13

Greg Kelly, Hanford High School, Richland, Washington

Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington. Disks and Washers: Using Integration to Find Volume. Limerick Nuclear Generating Station, Pottstown, Pennsylvania. Suppose I start with this curve.

vicky
Télécharger la présentation

Greg Kelly, Hanford High School, Richland, Washington

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Disks and Washers: Using Integration to Find Volume Limerick Nuclear Generating Station, Pottstown, Pennsylvania

  2. Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a cone in this shape. So I put a piece of wood in a lathe and turn it to a shape to match the curve. Find the volume of the solid created when the graph is rotated about the x-axis.

  3. 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. In this case: r= the y value of the function thickness = a small change in x =dx

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

  5. Example • Find the volume of the solid that results when the region enclosed by is revolved about the x-axis.

  6. Example • Find the volume of the solid that results when the region enclosed by is revolved about the y-axis

  7. 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.

  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.

  9. Example • Find the volume of the solid that results when the region enclosed by is revolved about the x-axis.

  10. Example • Find the volume of the solid that results when the region enclosed by is revolved about the y-axis.

  11. Example • Find the volume of the solid that results when the region enclosed by is revolved about the y-axis.

  12. Example • Find the volume of the solid that results when the region enclosed by is revolved about the line x = 2.

  13. Practice • Pg. 456 • 1, 3, 7, 9, 19, 23

More Related