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Thread: length of a curve

  1. #1
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    length of a curve

    Find the length of :

    1) $\displaystyle r(t) = cos(t){\bold{i}}+ sin(t){\bold{j}} + ln(cos(t)){\bold{k}}$

    2) $\displaystyle r(t) = {\bold{i}} + t^2{\bold{j}} + t^3{\bold{k}} \ \ \ 0 \leq t \leq 1$

    so far I have:
    1) $\displaystyle r'(t) = (-sin(t){\bold{i}} + cos(t){\bold{j}}+ tan(t){\bold{k}}$

    $\displaystyle |r'(t)| = \sqrt{sin^2(t) + cos^2(t) + tan^2(t)} = \sqrt{1+tan^2(t)} = \sqrt{sec^2(t)} = sec(t)$

    $\displaystyle \int^{\frac{\pi}{4}}_{0}sec(t) = \int^{\frac{\pi}{4}}_{0}sec(\frac{\pi}{4})$

    2) $\displaystyle r'(t) = (0 + 2t + 3t^2) \ \ \ |r(t)|= \sqrt{4t^2 + 9t^4} = t \sqrt{4 + 9t^2}$

    is 1) correct and for 2) I'm stuck.
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  2. #2
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    $\displaystyle \int{t\sqrt{4+9t^{2}}}dt$

    Let $\displaystyle u=4+9t^{2}, \;\ du=18tdt, \;\ \frac{1}{18}du=tdt$

    $\displaystyle \frac{1}{18}\int{u^{\frac{1}{2}}}du$

    Now, integrate and resub.
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