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Math Help - Find the length of the curve

  1. #1
    Junior Member
    Joined
    Nov 2009
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    Find the length of the curve

    I have tried this one over and over and can't get the right answer. Can anyone help?

    Consider the parametric equation
    x = 16(cosθ+ θsinθ)
    y = 16(sinθ− θcosθ)
    What is the length of the curve for θ = 0 to θ = [3/10] π?

    I have L=sqrt (dx/dt)^2 + (dy/dt)^2
    dx/dt = 16t (sin(t)+sin(t)+t cos(t))
    dy/dt= 16t (-cos(t)-cos(t)+t sin(t))

    Is that right?
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  2. #2
    Senior Member
    Joined
    Mar 2009
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    You will use the arclength formula
    L = \int_0^{\frac{3}{10}n} \! \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}\,dt.
    \frac{dx}{d\theta} = 16\left(-\sin \theta + \sin \theta + \theta \cos \theta\right) = 16\cdot \theta \cos \theta
    \frac{dy}{d\theta} = 16\left(\cos \theta - \cos \theta + \theta \sin \theta\right) = 16\cdot \theta \sin \theta
    Now substitute this into the arclength formula and remember \cos^2 \theta + \sin^2 \theta = 1.
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