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Math Help - please help me the problem using beta and gamma

  1. #1
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    Lightbulb please help me the problem using beta and gamma

    hi:
    have a problem solving this:
    solved by using beta and gamma function

    show that the area enclosed by the curve X^4+y^4=1
    is {Gamma(1/4) }^2/2(sqrt(pi))

    Could someone please help me

    tnanks .
    Last edited by wesam; January 18th 2010 at 04:10 AM.
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  2. #2
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    Quote Originally Posted by wesam View Post
    hi:
    have a problem solving this:
    solved by using beta and gamma function

    show that the area enclosed by the curve X^4+y^4=1
    is {Gamma(1/4) }^2/2(sqrt(pi))

    Could someone please help me

    tnanks .
    Well, one way to do that, since this is a closed curve, is to change to polar coordinates: let x= r cos(\theta) and [tex]y= r sin(\theta). Now the equation of the curve is [tex]r^4 cos^4(\theta)+ r^4 sin^4(\theta)= 1 or r= \frac{1}{\sqrt[4]{cos^4(\theta)+ sin^4(\theta)}} and then the area is given by
    \int_{\theta= 0}^{2\pi}\int_{r=0}^\frac{1}{\sqrt[4]{cos^4(\theta)+ sin^4(\theta)}} r drd\theta = \int_{\theta= 0}^{2\pi}\frac{1}{2(\sqrt[4]{cos^4(\theta)+ sin^4(\theta)})^2} d\theta = \int_{\theta= 0}^{2\pi}\frac{1}{2\sqrt{cos^4(\theta)+ sin^4(\theta)}} d\theta.
    .

    Now, what are the definitions of the Gamma and Beta functions and how do they relate to that integral?
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Well, one way to do that, since this is a closed curve, is to change to polar coordinates: let x= r cos(\theta) and [tex]y= r sin(\theta). Now the equation of the curve is [tex]r^4 cos^4(\theta)+ r^4 sin^4(\theta)= 1 or r= \frac{1}{\sqrt[4]{cos^4(\theta)+ sin^4(\theta)}} and then the area is given by
    \int_{\theta= 0}^{2\pi}\int_{r=0}^\frac{1}{\sqrt[4]{cos^4(\theta)+ sin^4(\theta)}} r drd\theta = \int_{\theta= 0}^{2\pi}\frac{1}{2(\sqrt[4]{cos^4(\theta)+ sin^4(\theta)})^2} d\theta = \int_{\theta= 0}^{2\pi}\frac{1}{2\sqrt{cos^4(\theta)+ sin^4(\theta)}} d\theta.
    .

    Now, what are the definitions of the Gamma and Beta functions and how do they relate to that integral?
    thank you very much
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