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Math Help - calc 3- line integral

  1. #1
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    Exclamation calc 3- line integral

    find the mass of a wire in the shape of the helix traced by (cost, sint, t/pi)
    from pi to 3 pi if its density at each point is proportional to the distance from the point to the xy-plane

    so i have mass= the integral from pi to 3pi of
    cost + sint + t/pi dL

    but i'm not sure what dL is..i saw it in an example in the book.
    is dL the derivative of what i have above? or is it the magnitude? or do i ignore it?
    after i clarify this, i think i'll be able to hopefully solve it..
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    dL means that the integral is with respect to arc length. Imagine if the wire had a uniform mass density - then its mass would clearly be equal to its length times the mass density.

    You probably know that the length of a parametrized curve \phi between \phi(a) and \phi(b) is

    L=\int_a^b|\phi'(s)|ds

    so dL = |\phi'(s)|ds. Therefore the integral you are looking for is \int_{\pi}^{3\pi}W(\phi(s))|\phi'(s)|ds where W(\phi(s)) is the mass density of the wire at the point \phi(s). You are told that this is proportional to the distance from the point \phi(s) to the XY plane, i.e. W(\phi(s))=k \times d(\phi(s), XY\mbox{ plane}) for some constant k. Find the expression for that, substitute it into the integral and then evaluate it.
    Last edited by Bruno J.; October 27th 2009 at 04:42 PM.
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  3. #3
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    i'm sorry i'm sort of confused with all of your symbols. my final answer is:
    4*sqrt(pi^2 +1)
    is that right/close?
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    I didn't do the calculation! Please post your work, and I'll tell you if it's good.
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  5. #5
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    f(x)= cost+sint+t/pi
    dL= -sint+cost+1/pi then i took the magnitude of this to get
    dL= (sqrt.(pi^2+1))/(pi)

    so i took the integral from pi to 3pi of f(x)*dL
    and got

    (sint-cost+(1/2pi)*t^2)*(sqrt(pi^2 +1))/(pi))* t

    and evaluated from 3pi to pi
    and got

    a final answer of

    18pi* sqrt(pi^2 +1)

    i found a few algebraic answers and fixed my first original answer to get this
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  6. #6
    MHF Contributor Bruno J.'s Avatar
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    Well the function isn't f(t)=\cos t+\sin t+\frac{t}{\pi}, it's f(t) = (\cos t, \sin t, \frac{t}{\pi}).

    Moreover, what did you do of the fact that the mass density at the point f(t) is proportional to the distance between f(t) and the XY plane? If you do not take that into account in your solution then your solution is certainly wrong.
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