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Math Help - Unit square integral

  1. #1
    Jem
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    Unit square integral

    Compute the following double integrals taken on an unit square:

    • \int_0^1\int_0^1\sqrt{x^2+y^2}\,dx\,dy
    • \int_0^1\int_0^1\frac{dx\,dy}{\sqrt{1+x^2+y^2}}
    • \int_0^1\int_0^1\frac{dx\,dy}{1-xy}
    • \int_0^1\int_0^1\frac{dx\,dy}{(2-xy)\ln(xy)}
    • \int_0^1\int_0^1\frac{1-x}{\{-\ln(xy)\}^{5/2}}\,dx\,dy
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  2. #2
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    Quote Originally Posted by Jem View Post
    [*] \int_0^1\int_0^1\frac{dx\,dy}{1-xy}
    \int_0^1 \int_0^1 \frac{dx~dy}{1-xy} = \int_0^1 \int_0^1 \sum_{n=0}^{\infty} x^ny^n ~ dx~dy = \sum_{n=0}^{\infty} \frac{1}{(n+1)^2} = \frac{\pi^2}{6}.

    See This.
    Last edited by ThePerfectHacker; February 10th 2008 at 08:03 AM.
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  3. #3
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    Quote Originally Posted by Jem View Post
    • \int_0^1\int_0^1\frac{dx\,dy}{1-xy}
    This one can also be killed by setting (x,y)=(u-v,u+v). From there you compute the Jacobian and make a sketch of the new region to split the new square into two triangles. Finally you get the famous result \frac{\pi^2}6 which is the Basel problem.
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