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Math Help - Arclength?

  1. #1
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    Arclength?

    The problem states, r(t)=(6t,3t^2,3log(t)), find the length from the points (6,3,0) and (24,48,3log(4)). I thought it would be the integral from 1-4 of the square root of the derivatives of each squared, added together. Seems that this isn't correct.

    My derivatives are r'(t)=(6,6t,3/(tln10))

    My final answer is 48.928761 but this was NOT correct. Can someone offer some help here?
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  2. #2
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    Quote Originally Posted by Bracketology View Post
    The problem states, r(t)=(6t,3t^2,3log(t)), find the length from the points (6,3,0) and (24,48,3log(4)). I thought it would be the integral from 1-4 of the square root of the derivatives of each squared, added together. Seems that this isn't correct.

    My derivatives are r'(t)=(6,6t,3/(tln10))

    My final answer is 48.928761 but this was NOT correct. Can someone offer some help here?


    I get \displaystyle{\int\limits^4_1\sqrt{6^2+6^2t^2+\fra  c{3^2}{t^2}}\,dt=3\int\limits^4_1\frac{2t^2+1}{t}\  dt=45+3\ln 4 , which

    is slightly more that what you got.

    Tonio
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  3. #3
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    Where did you get the 3^2/t^2? I thought it was 3/tln10, that's where Mine was different. Your answer is correct.
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  4. #4
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    Quote Originally Posted by Bracketology View Post
    Where did you get the 3^2/t^2? I thought it was 3/tln10, that's where Mine was different. Your answer is correct.
    the derivative of 3\log{t} is \dfrac{3}{t} ... usually in more advanced math texts, "log" is understood to be base e, not base 10.
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    Quote Originally Posted by Bracketology View Post
    Where did you get the 3^2/t^2? I thought it was 3/tln10, that's where Mine was different. Your answer is correct.

    This may be confussing: though we're taught in high school, and thus we're used, to

    take \log=\log_{10}\,,\,\,and\,\,\ln=\log_e (as it's usual in conventional hand calculators), in

    fact many mathematicians see both log and ln as one and the same.

    In fact, when I saw your question that's the first thing I thought, namely: that we have

    here log = the natural logarithm (i.e., ln) and not logarithm to base 10.

    This was the whole problem...

    Tonio
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