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Integral of function over ellipse

  1. Feb 27, 2013 #1

    I'm trying to find
    [tex] \iint_S \sqrt{1-\left(\frac{x}{a}\right)^2 -\left(\frac{y}{b}\right)^2} dS [/tex]

    where S is the surface of an ellipse with boundary given by [itex]\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1 [/itex].

    Any suggestions are appreciated!


  2. jcsd
  3. Feb 27, 2013 #2
    Do you mean the interior of an ellipse?

    Anyway, the first thing I though of is Green's Theorem for some reason. Probably since then we can make the substitution ##\left(\dfrac xa\right)^2+\left(\dfrac yb\right)^2=1##.

    The second thing I thought of was a change of coordinates and a multiplication by the Jacobian determinant, then we have it reduced to

    $$a\cdot b\cdot\iint_C\sqrt{1-m^2-n^2}\mathrm{d}S'$$

    where C is the unit circle wrt m and n and S' should have a fairly obvious definition.
    Last edited: Feb 27, 2013
  4. Feb 27, 2013 #3


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    Try the substitution$$
    \frac x a = r\cos\theta,\,\frac y b = r\sin\theta$$
  5. Feb 27, 2013 #4
    Got it!

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