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Math Help - Total length of the astroid

  1. #1
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    Total length of the astroid

    This is the last section of my class and I just don't get it. I can't even get my calculator so graph it for me (probably my stupid though). Can anyone please explain how to answer this type of question? I have 5 questions on the length of a parametric curve to do. I just can't understand what the book is saying (again probably my stupid). Any help is greatly appreciated.

    If f(θ) is given by: f(θ)=18cos3θ and g(θ) is given by: g(θ)=18sin3θ
    Find the total length of the astroid described by f(θ) and g(θ).
    (The astroid is the curve swept out by (f(θ),g(θ)) as θ ranges from 0 to 2π. )
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  2. #2
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    f(t)=18cos(3t), \;\ g(t)=18sin(3t)

    f'(t)=-54sin(3t), \;\ g'(t)=54cos(3t)

    parametric arc length is given by \int_{0}^{2\pi}\sqrt{(-54sin(3t))^{2}+(54cos(3t))^{2}}dt=54\int_{0}^{2\pi  }dt

    I suppose this is what they are getting at. To graph this in parametric, your calculator must be in parametric mode. Parametrically, this is an ellipse, not an astroid. Unless I am missing something. The convention normally used is t instead if theta. Theta is commonly used when dealing with polar coordinates. Even in polar coordinates, this is a rose and not an astroid.

    Please do not tell me you mean f(t)=18cos^{3}(t), \;\ g(t)=18sin^{3}(t) and did not use ^ to mean power.
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  3. #3
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    That is what the equation reads. All greek to me . Thanks for your help.
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  4. #4
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    For an astroid, the equations should be x(t)=18cos^{3}(t) \;\ and \;\ y(t)=18sin^{3}(t).

    x'(t)=-54sin(t)cos^{2}(t)

    y'(t)=54sin^{2}(t)cos(t)

    Repeating, Parametric Arc length is give by \int_{a}^{b}\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{d  t})^{2}}dt

    Yours whittles down to

    54\int_{0}^{2\pi}|sin(t)cos(t)|dt

    Let's do it this way though. Since we have symmetry, we can integrate from 0 to Pi/2 and multiply by 4:

    216\int_{0}^{\frac{\pi}{2}}sin(t)cos(t)dt

    Can you integrate that?. See the graph?. That is an astroid. Astroid means star-shaped.
    Attached Thumbnails Attached Thumbnails Total length of the astroid-astroid.jpg  
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