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Finding Volume of an Elipsoid

  1. Nov 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the volume of Elipsoid x^2+y^2+5z^2=16


    3. The attempt at a solution
    So if x and y are both zero z goes from -(16/5)^(1/2) to (16/5)^(1/2)
    and if I do it in polar coordinates then r goes from 0 to 4
    and theta goes from 0 to 2pi?

    so Triple Integral: rdrdθdz with the above parameters should give me the correct answer... right?
     
  2. jcsd
  3. Nov 5, 2013 #2

    vanhees71

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    2016 Award

    How about parametrizing the ellipsoid as
    [tex]\vec{r}(\lambda,\theta,\phi)=\lambda \begin{pmatrix}
    a \cos \phi \sin \theta \\
    b \sin \phi \sin \theta \\
    c \cos \theta
    \end{pmatrix}?
    [/tex]
    Now you have to find the boundaries of the three parameters and check that it's really giving the ellipsoide. Then evaluate the Jacobian and do the integral :-).
     
  4. Nov 5, 2013 #3
    ah is there a way to do it with cylindrical coordinates like the way I was doing it? I think the problem wants me to use this method X_x.
     
  5. Nov 5, 2013 #4

    haruspex

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    Yes, but z and r are not independent. The value of one will affect the bound on the other.
    (Far the easiest way here is to substitute t = z*constant, if you pick the right constant.)
     
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