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Thread: Length of a Curve

  1. #1
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    Length of a Curve

    Consider x=(e^t)+(e^-t), y=5-2t, 0<t<3. Find the length of the curve.
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  2. #2
    Member Mentia's Avatar
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    Recall the arc length integral:

    $\displaystyle = \int_{ {t }_{i } }^{ {t }_{f } } \left | r'(t) \right | dt$

    Where,

    $\displaystyle r'(t) = \sqrt[ ]{x'(t)^{2}+y'(t)^{2} } $

    So we have:

    $\displaystyle x(t) = e^{t}+e^{-t} = 2cosh(t)$
    $\displaystyle y(t) = 5 - 2t$.

    Then,

    $\displaystyle x'(t) = 2sinh(t)$
    $\displaystyle y'(t) = -2$.

    So,

    $\displaystyle r'(t) = \sqrt[ ]{4 \sinh ^2(t) + 4} = 2 \cosh (t)$.

    Then,

    Arc Length = $\displaystyle \int_{0}^{3} 2 \cosh (t) = 2 \sinh (3) - 2 \sinh (0) = 2 \sinh (3) \approx 20.04 $
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  3. #3
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    ok i get it but how did you get the 2cosht from (e^t)-(e^-t)?
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  4. #4
    Member Mentia's Avatar
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    Check out hyperbolic cosine from wikipedia: Hyperbolic function - Wikipedia, the free encyclopedia
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