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Volumes formed by parabolas

Kevin Malone

on 16 April 2010

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Transcript of EDT430

Volumes Formed by Parabolas Consider the parabola above. What shape might be created if the shaded area were rotated about the x-axis? Now what if this parabola were rotated about the y-axis? Let's start with rotations about the x-axis... Things to remember:

How did we approximate the area under a parabola?
How did we calculate the volume of a frustum? What do these rotated sections look like? Let's check it out. What is R?
What is r?
What is h? We can approximate the volume created by the entire figure by adding the volumes of several frustums. Do you see this? Let's look at an example... A B C D E F Total Volume = Vol A + Vol B + ... + Vol F Summation Notation It's cumbersome to write this entire equation out, so we can shorten it using summation notation or in general, Now for rotations about the y-axis What is a?
What is b?
What is h? When we rotate about the y-axis, the volume created by one section (say, the pink one), is the difference of two frustums. What are these two frustums? How could we calculate the volume of this parabola rotated about the y-axis, if we know that the volume is the difference of two frustums? In groups of 2-3, work to derive a formula to find this volume. Can you find a general formula? http://mathplotter.lawrenceville.org/mathplotter/mathPage/index.htm
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