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Math Help - Finding equation for a curve, given length integral

  1. #1
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    Finding equation for a curve, given length integral

    Hey,

    So I'm a bit confused on how an answer was reached. The question states:

    Find a curve through the point (1,1) whose length integral is given by:

    L = \int_1^4 \sqrt{1+\frac{1}{4x}}dx. From here I try to use point-slope formula, where m = \frac{1}{2\sqrt{x}} (which comes from taking the square root of \frac{1}{4x}. Now, I apply the point-slope formula to try and get the equation of the curve, using:

    y - y1 = m(x-x1) ==> y - (1) = \frac{1}{2\sqrt{x}} (x - (1))

    After adding 1 to both sides and distributing the \frac{1}{2\sqrt{x}} to the (x - 1), I end up with something like:

    y = \frac{x+2\sqrt{x}-1}{2\sqrt{x}}... which isn't anything like the y = \sqrt{x} answer it SHOULD be.

    Can someone point out where I went wrong?
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  2. #2
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    Re: Finding equation for a curve, given length integral

    Quote Originally Posted by Calcme View Post
    Hey,

    So I'm a bit confused on how an answer was reached. The question states:

    Find a curve through the point (1,1) whose length integral is given by:

    L = \int_1^4 \sqrt{1+\frac{1}{4x}}dx. From here I try to use point-slope formula, where m = \frac{1}{2\sqrt{x}} (which comes from taking the square root of \frac{1}{4x}. Now, I apply the point-slope formula to try and get the equation of the curve, using:

    y - y1 = m(x-x1) ==> y - (1) = \frac{1}{2\sqrt{x}} (x - (1))

    After adding 1 to both sides and distributing the \frac{1}{2\sqrt{x}} to the (x - 1), I end up with something like:

    y = \frac{x+2\sqrt{x}-1}{2\sqrt{x}}... which isn't anything like the y = \sqrt{x} answer it SHOULD be.

    Can someone point out where I went wrong?
    1. The length of a curve with the equation y = f(x) is calculated by:

    L=\int(\sqrt{1+(y')^2})dx

    2. Therefore (y')^2=\frac1{4x}~\implies~y' = \sqrt{\frac1{4x}} = \frac1{2\sqrt{x}}

    Thus y = \int\left(\frac1{2\sqrt{x}}  \right)dx = \sqrt{x} + c
    Last edited by earboth; July 25th 2011 at 09:24 PM. Reason: typo
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  3. #3
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    Re: Finding equation for a curve, given length integral

    Ahh, I see... we're just working all the way backwards until we get our original curve/function.

    Thanks a lot.
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