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Math Help - The length of a curve with polar coordinates

  1. #1
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    The length of a curve with polar coordinates

    [Edited below] see the third post for a more detailed attempt to solve it

    Hi guys,

    Problem: A curve is given in polar coordinates by r = fi^2 between -pi <= fi =< pi

    Determine the length of the curve.

    My progress:

    I recall a formula my teacher has given us: L(x) = $(using given limits of integration) SQRT (h^2(fi)) + [h'(fi)]^2) d(fi)

    = This would give me

    $SQRT (fi^2)^2 + (2fi)^2 d(fi)

    So far so good, right?

    $SQRT (fi)^4 + 4fi^2 d(fi)

    I think Variable substititution would be good here

    U = (fi)^4 + 4fi^2 d(fi)

    DU/D(fi) = 4(fi)^3 + 8fi

    Hmm How can i complete this?
    could someone help me in detail, thanks



    Last edited by Riazy; February 16th 2011 at 09:54 PM.
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  2. #2
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    Quote Originally Posted by Riazy View Post
    Hi guys,

    Problem: A curve is given in polar coordinates by r = fi^2 between -pi <= fi =< pi

    Determine the length of the curve.

    My progress:

    I recall a formula my teacher has given us: L(x) = $(using given limits of integration) SQRT (h^2(fi)) + [h'(fi)]^2) d(fi)

    = This would give me

    $SQRT (fi^2)^2 + (2fi)^2 d(fi)

    So far so good, right?

    $SQRT (fi)^4 + 4fi^2 d(fi)

    I think Variable substititution would be good here

    U = (fi)^4 + 4fi^2 d(fi)

    DU/D(fi) = 4(fi)^3 + 8fi

    Hmm How can i complete this?
    could someone help me in detail, thanks


    If I am reading this correctly you have

    r=\phi^2

    The arc length formula in polar coordinates is

    \displaystyle s=\int_{-\pi}^{\pi}\sqrt{r^2+\left(\frac{dr}{d\phi} \right)^2}d\phi

    Plugging in gives

    \displaystyle \int_{-\pi}^{\pi}\sqrt{(\phi^2)^2+(2\phi)^2}d\phi=\int_{a  }^{b}\phi\sqrt{\phi^2+4}d\phi
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  3. #3
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    Hi again guys I would like if someone here could explain the steps in the following problem:

    Problem:

    A curve is given by polar coordinates by r= (fi)^2, between -pi <= fi <= pi
    Determine the length of the curve.

    Solution ( I have got it from a friend, but don't understand all of it)
    // "I am not able to get in touch with him//

    1. We know that r = f(fi) = fi^2, and that its between -pi <= fi <= pi.

    2. We can write this as r^2 + ('r)^2 = fi^4 +(2fi)^2 = fi^4 + 4fi^2 = fi^2(fi^2+4)

    3. ds = SQRT [r^2+(r')^2] dfi = I(Fi)I (<--- Absolute value of fi, by taking SQRT) ->
    I(Fi)I SQRT[fi^2 +f] dfi -> S = $(limits of integration from - pi to pi)I(Fi)I SQRT[fi^2+4] dfi

    4 = 2 $ (from 0 to pi) SQRT[fi^2 + 4fi] dfi = /u= fi^2 + 4 =-> du = 2fi(dfi) / =

    5. $(limits of integration from 4 to 4+pi^2) SQRT[u]du = [2/3 * u* SQRT [u] ] (from 4 to pi + 4)

    = 2/3 (( pi^2 + 4)^3/2 -8)


    The thing is that I find many of the steps confusing and I don't understand what has really been done


    for example
    Step 4: Where does the constant 2 come from?, how did the limits of integration change into 0-pi
    Step5: Limits of 4 to 4+pi^2

    I would like if someone could make those to steps in a little more detail,

    Thank you guys.
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