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Volumes of Solids of Revolution: Disc Method

Volumes of Solids of Revolution: Disc Method. Adapted by Mrs. King from http://tutorial.math.lamar.edu/AllBrowsers/2413/VolumeWithRings.asp. Volume of a Cylinder:. Remember Riemann Sums?. Why did we use these? Where did the process lead us?. What is a solid of revolution?.

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Volumes of Solids of Revolution: Disc Method

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  1. Volumes of Solids of Revolution: Disc Method Adapted by Mrs. King from http://tutorial.math.lamar.edu/AllBrowsers/2413/VolumeWithRings.asp

  2. Volume of a Cylinder:

  3. Remember Riemann Sums? Why did we use these? Where did the process lead us?

  4. What is a solid of revolution? • In this section we will study the volume of a solid of revolution.  • To get a solid of revolution we start out with a function, y=f(x), on an interval [a,b].

  5. What is a Solid of Revolution Consider the area under the graph of y = 0.5x from x = 0 to x = 1:

  6. What is a Solid of Revolution 2 If the shaded area is now rotated about the x-axis, then a three-dimensional solid (called Solid of Revolution) will be formed: What will it look like? Pictures from http://chuwm2.tripod.com/revolution/

  7. What will it look like? http://www.worldofgramophones.com/ victor-victrola-gramophone-II.jpg Consider the solid of revolution formed by the graph of y = x2 from x = 0 to x = 2: How is it calculated

  8. Visual Explanation • To find the volume, we are “slicing” the solid into n disc, then adding the volumes of these discs together. • How can we improve the accuracy of our estimate? http://library.thinkquest.org/3616/Calc/S3/TDM.html

  9. Discs • Once again, we need to be able to add a collection of infinite items! f(x) a b

  10. Example • Determine the volume of the solid obtained by rotating the region bounded by , , and the x-axis about the x-axis. • First, graph the function.

  11. Homework • Page 465 • #1-4, 11a, 12b, 25, 26*

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