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Math Help - fat curves in parametrics

  1. #1
    Junior Member
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    Jan 2010
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    fat curves in parametrics

    The question: The curves with equations x" + y" = 1, n = 4, 6, 8, , are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length S(2k), the arc of the fat circle with n=2k. Without attempting to evaluate the integral, state the value of lim S(2k) as x approaches infinity.


    I know we can use the Formula (for the lengths of curves.
    L= intergal of square root ((dx/dt)^2)/(dy/dt)^2) dt) to get the length

    when n=2, it is just a regular circle, therefore it is not too difficult to parametrize the function (for example, x=cost y=cost)
    However, when n gets other even number integers, i do not know how i can define it in parametrics.
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  2. #2
    Senior Member
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    Jan 2010
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    I think this will work:

    Let y=\sqrt[n]{1-x^n} (note that this is actually only the positive half (y>0) of the "fat circle", so you need to multiply your solution by 2 to get the length of the entire "fat circle")

    and then evaluate the length using the regular \int \sqrt{1+(y')^2} \, dx formula.
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