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Math Help - Implicit differentiation to find the length of a spiral

  1. #1
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    Implicit differentiation to find the length of a spiral

    Hey,
    I was wondering if there was a quicker/better way of working this out:

    x(t)=(t+a)cos(t), y(t)=(t+a)sin(t)
    therefore dx/dt= -(t+a)sin(t) + acos(t)
    and dy/dt= (t+a)cos(t) + asin(t)

    To work out the length I would have to use L=int{sqrt(1+ [dy/dx]^2 )}
    EDIT: forgot about the integral

    [dy/dx]^2 is a nightmare to work out
    do I have to crunch the numbers or is there a way to simplify and do it quicker?

    Thanks in advance

    EDIT: Oh.. and this is called parametric differentiation :S
    Last edited by fatlucky; January 9th 2011 at 07:41 AM. Reason: correction
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  2. #2
    MHF Contributor alexmahone's Avatar
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    s=\int_{t_1}^{t_2}\sqrt{(\frac{dx}{dt})^2+(\frac{d  y}{dt})^2}dt=\int_{t_1}^{t_2}\sqrt{(t+a)^2+a^2}dt
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  3. #3
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    Quote Originally Posted by alexmahone View Post
    s=\int\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt
    Thanks
    A useful one to remember
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