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Deducing the volume of an elliptical cone

  1. Jan 25, 2012 #1
    Here again

    1. The problem statement, all variables and given/known data

    Find the volume of a right elliptical cone with an elliptic base with semi-axes a and b and heigh h

    2. Relevant equations

    So: [itex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/itex]

    3. The attempt at a solution

    image.jpg

    image.jpg

    That's what I have, but answer should be:

    [itex]V=\frac{1}{3}abh\pi[/itex]

    I've checked it all over again like 10 times, but I can't find the mistake. If you can see it I'd be grateful
     
  2. jcsd
  3. Jan 25, 2012 #2

    LCKurtz

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    To tell you the truth, I can 't bring myself to slog through all your steps. But I have a question for you. Have you had change of variables in double integrals yet? For example, do you know how to show the area of the ellipse $$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1$$ is ##\pi a b## by mapping the ellipse to a circle? The reason for my asking is that if you have studied that, a much easier way to do the problem is to use the elliptical cross-sections. You can figure out the equation of the ellipse cross section at height ##z## for ##0\le z \le h## and either use the above formula or develop it with the appropriate change of variables.
     
  4. Jan 25, 2012 #3
    I know how to make an implicit differentation but, I can't do that to an intregal expression.
     
  5. Jan 25, 2012 #4

    LCKurtz

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    What I asked you has nothing to do with implicit differentiation. Just tell me this: Are you allowed to use the area formula ##\pi ab## for your standard xy ellipse area as a given? If the answer to that is yes, then figure out the equation of the elliptical cross section of your cone at height ##z## for ##0\le z \le h## and use that formula for its area. Then you can integrate the elliptical cross section area as a function of z to get the volume.
     
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