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General Equations for Certain Volumes of Revolution

  1. Jun 23, 2009 #1
    The volume of a cone =
    - B H where B is the base of the cone and H is its height.

    We can think about a cone as the line y = x rotated with respect to the y axis. The volume of a parabaloid =
    - B H

    and a parabaloid is the line y = x^2 rotated with respect to the y axis. So what if you wanted to know the what line has the equation 1/4*B*H or in general K*B*H, where K is a constant.
    Let y= a*x^n and y=H, which is the height. Then x = (y/a)^1/n.

    Then the integral for the volume of revolution becomes
    [0]\int[/H](y/a)^1/n dy

    Using a substitution and integrating we get that Volume= a*pi*((H/a)^((2+n)/n))/((2+n)/n)
    We know that V = K*pi*x^2*H, and therefore we can equate the two equations for volume, and using the relationship that H= ax^n we get

    a*pi*((H/a)^((2+n)/n))/((2+n)/n) = K*a*pi*x^2*x^n

    Pi and a cancel out on both sides. Using the relationships x^2 =(y/a)^2/n and y=h we get

    ((H/a)^((2+n)/n))/((2+n)/n))= K*((H/a)^((2+n)/n))

    and ((H/a)^((2+n)/n)) cancels out on both sides leaving

    K= n/(n+2) where n is the value to which x is raised: x^n. Solving for n we get that n=2k/(1-k). Since we can think of a cone as the line y = x rotated with respect to the y-axis, then we need to let n=1 and see if k=1/3, which it does. You could also use this to determine what shape has the equation 1/4B*H by letting k=1/4, and you get n=2/3. So the line y= x^(2/3) rotated with respect to the y-axis has the equation 1/4*B*H

    Having done all of this work, does anyone have any suggestions on where to go from here with this work?
  2. jcsd
  3. Jun 24, 2009 #2
    Sorry, this is not the answer to your question, but the forum provides a really nice latex display (enclose it in "tex" tags) that would make your equations MUCH easier to read :)
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