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Math Help - Calculate the surface area of the surface

  1. #1
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    Calculate the surface area of the surface

    Calculate the surface area of the part of the sphere x^2 + y^2 + z^2 = 1 that is contained in the lemniscata (x^2 + y^2)^2 = x^2 - y^2

    I solved this quickly but i cant seem to get the same answer given in the book, And i would like to know which is the correct one. thanks in advance to anyone that helps.
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  2. #2
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    Quote Originally Posted by karpatzio View Post
    Calculate the surface area of the part of the sphere x^2 + y^2 + z^2 = 1 that is contained in the lemniscata (x^2 + y^2)^2 = x^2 - y^2

    I solved this quickly but i cant seem to get the same answer given in the book, And i would like to know which is the correct one. thanks in advance to anyone that helps.
    Show us what you did.
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  3. #3
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    the area element of a sphere given in x,y is \sqrt{\dfrac{1}{1-x^2-y^2}}dxdy
    integrating it over the are of the lemniscata:

    \int\int \sqrt{\dfrac{1}{1-x^2-y^2}}dxdy
    D

    when i change to polar coordinates
    x = r*cos(\phi)
    y = r*sin(\phi)

    i get 4\displaystyle\int^{\pi/4}_{-\pi/4} \, d\phi\displaystyle\int^{\sqrt{cos(2\phi)}}_0 \dfrac{r}{1-r^2}\, dr = -4\displaystyle\int^{\pi/4}_{-\pi/4} \sqrt{1 - cos(2\phi)} - 1\, d\phi =

    = 4*(\sqrt{2}*cot(\phi)*\sqrt{sin^2(\phi)} + \phi [\pi/4 , -\pi/4])

    eventually solving and getting the wrong answer, what am i doing wrong?
    Last edited by karpatzio; June 26th 2009 at 04:37 AM.
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