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Math Help - Integral Question

  1. #1
    Member Em Yeu Anh's Avatar
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    Question Integral Question

    Two parts to this problem:
    1. Show that the area of the ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 is {\pi}ab;

    2. Find the volume enclosed by the ellipsoid \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 by integrating the area of a horizontal cross section.

    I can't find any similar examples in my textbook and I am really stumped on this one, I'm taking a guess that this is meant to be solved by double integrals? If so, I am terrible at setting those up >.< If someone could point me in the right direction for solving it would help a lot =)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Em Yeu Anh View Post
    Two parts to this problem:
    1. Show that the area of the ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 is {\pi}ab;
    Parameterize the ellipse by x=a\cos(\theta),y=b\sin(\theta) and remember the area for a parametric curve is \int_a^b x'(t)y(t)dt. From this we see that ab\int_0^{2\pi}\sin^2(\theta)d\theta=\pi ab

    Follow suit for the second one.

    EDIT: Typo.
    Last edited by Drexel28; January 5th 2010 at 09:26 PM.
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  3. #3
    Member Em Yeu Anh's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Parameterize the ellipse by x=a\cos(\theta),y=b\sin(\theta) and remember the area for a parametric curve is \int_a^b x'(t)y(t)dt. From this we see that ab\int_0^{2\pi}\sin^2(\theta)d\theta=\pi ab

    Follow suit for the second one.

    EDIT: Typo.
    Thanks a bunch.
    Would you mind explaining this part to me, I'm slightly confused about this:
    so x'(t) = -asint is that correct?
    Would the area for the parametric curve then be -ab\int_0^{2{\pi}}sin^2{\theta}d{\theta} ?
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