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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.

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Greg Kelly, Hanford High School, Richland, Washington

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  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

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