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Greg Kelly, Hanford High School, Richland, Washington

Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington. Discs. Limerick Nuclear Generating Station, Pottstown, Pennsylvania. Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.

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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 Discs 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 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.

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

  5. This application of the method of slicing is called the disc 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. If the shape is rotated about the x-axis, then the formula is: A shape rotated about the y-axis would be: Since we will be using the disc method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.

  6. 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 . We use a horizontal disc. The thickness is dy. volume of disk

  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 disc.

  8. 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 “disc” now has a hole in it, making it a “washer”. outer radius inner radius

  9. 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 disc method, this formula will not be on the formula quizzes. I want you to understand the formula.

  10. r R 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. We can use the washer method if we split it into two parts: cylinder inner radius outer radius thickness of slice

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