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Math Help - Question regarding the equation of an ellipsoid

  1. #1
    Member Em Yeu Anh's Avatar
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    Angry Question regarding the equation of an ellipsoid

    Ellipsoid: \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1

    Quick question. I am to take a horizontal cross-section of this shape and use it to find the volume of the entire shape by the slicing method. Area of an ellipse is given, {\pi}ab. These are the steps that my prof wrote for me:

    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1-\frac{z^2}{c^2}

    \frac{x^2}{a^2(1-\frac{z^2}{c^2})}+\frac{y^2}{b^2(1-\frac{z^2}{c^2})} = 1

    A(z) = {\pi}(a\sqrt{1-\frac{z^2}{c^2}})(b\sqrt{1-\frac{z^2}{c^2}})

    So V = {\pi}ab\int_{z=-c}^{z=c}(1-\frac{z^2}{c^2})dz

    When integrated I did obtain the answer of \frac{4{\pi}abc}{3} but I am just not sure how to correctly determine the limits of integration (z=-c to z=c). Thanks!
    Last edited by Em Yeu Anh; January 11th 2010 at 05:57 PM. Reason: LaTex mistakes
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  2. #2
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    Quote Originally Posted by Em Yeu Anh View Post
    Ellipsoid: \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1

    Quick question. I am to take a horizontal cross-section of this shape and use it to find the volume of the entire shape by the slicing method. Area of an ellipse is given, {\pi}ab. These are the steps that my prof wrote for me:

    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1-\frac{z^2}{c^2}

    \frac{x^2}{a^2(1-\frac{z^2}{c^2})}+\frac{y^2}{b^2(1-\frac{z^2}{c^2})} = 1

    A(z) = {\pi}(a\sqrt{1-\frac{z^2}{c^2}})(b\sqrt{1-\frac{z^2}{c^2}})

    So V = {\pi}ab\int_{z=-c}^{z=c}(1-\frac{z^2}{c^2})dz

    When integrated I did obtain the answer of \frac{4{\pi}abc}{3} but I am just not sure how to correctly determine the limits of integration (z=-c to z=c). Thanks!
    You want to integrate over the entire ellipsoid. The ellipsoid itself extends from z= -c up to z= c.
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