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Math Help - Length of a curve?

  1. #1
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    Length of a curve?

    I have the function f(x) = x^2, is there a way to measure the length of the line between (0,0) and (2,4) ?
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  2. #2
    MHF Contributor Quick's Avatar
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    Specify

    Are you trying to find the distance between the two points or the length of the curved line along the two points?
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  3. #3
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    Quote Originally Posted by Quick
    Are you trying to find the distance between the two points or the length of the curved line along the two points?
    The distance of the curved line
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  4. #4
    TD!
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    Yes, that's possible with integration.

    The arc length of a function f(x) between x = a and x = b is given by:

    \ell  = \int\limits_a^b {\sqrt {1 + f'\left( x \right)^2 } dx}

    In your case (I omitted the calculation), the result is:

    <br />
\ell  = \int\limits_0^2 {\sqrt {1 + 4x^2 } dx}  = \frac{1}{4}\ln \left( {\sqrt {17}  + 4} \right) + \sqrt {17}  \approx 4.647<br />
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  5. #5
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    Quote Originally Posted by TD!
    Yes, that's possible with integration.

    The arc length of a function f(x) between x = a and x = b is given by:

    \ell  = \int\limits_a^b {\sqrt {1 + f'\left( x \right)^2 } dx}

    In your case (I omitted the calculation), the result is:

    <br />
\ell  = \int\limits_0^2 {\sqrt {1 + 4x^2 } dx}  = \frac{1}{4}\ln \left( {\sqrt {17}  + 4} \right) + \sqrt {17}  \approx 4.647<br />
    I havn't started integration yet, (still studying calculus) can you please explain how you derived the formula and how you evaluated it?

    Thanks
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  6. #6
    TD!
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    Quote Originally Posted by chancey
    I havn't started integration yet, (still studying calculus) can you please explain how you derived the formula and how you evaluated it?

    Thanks
    If you haven't seen integration yet, you won't understand much of a short explanation. The evaluation isn't that easy either, at least not for a beginner.

    Perhaps you could do some reading: here and/or here.
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  7. #7
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    OK thanks, I'll give it a read
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by chancey
    I havn't started integration yet, (still studying calculus)
    Integration is part of calculus, you may still be studying differentiantion.

    RonL
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