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Arc Length Problem

  1. Oct 15, 2009 #1
    Hey guys,

    Got a bit of a problem with a question I found in a text book. I can do most of it but theres one little part I'm really struggling with:

    A curve C is given parametrically by:
    x=t-tanht, y=secht, t[tex]\geq0[/tex]​

    The length of arc C measured from the point (0,1) to a general point with parameter t is s. Find s in terms of t and deduce that, for any point on the curve, y=e-s.


    I'm happy finding that the arc length is defined as [tex]\int (tanht)dt[/tex] between the limits of 0 and s, and i evaluate this integral to be ln(coshs) however after this I am stumped; I am having great trouble getting to y=e-s.


    Can anyone please help me out?


    Oscar
     
    Last edited: Oct 15, 2009
  2. jcsd
  3. Oct 15, 2009 #2

    LCKurtz

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    Perhaps you have misunderstood or mis-quoted the problem. Your work looks correct but the reason you can't get to y = e-s is the two aren't equal, as trying s = 0 will show.
     
  4. Oct 15, 2009 #3
    It is a past examination question and I quoted it word for word from what is in my text book. I've doubled checked it and what I have written is definitely what is written down here... so unless there is a typo in the textbook I really don't know what is going on :S


    Thanks for the speedy reply,
    Oscar
     
  5. Oct 15, 2009 #4

    LCKurtz

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    On looking again, perhaps the difficulty is that the problem is asking for y in terms of s, apparently not what you have calculated.
     
  6. Oct 15, 2009 #5

    LCKurtz

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    In fact it's trivial from the equations y = sech(t) and s = ln(cosh(t)).
     
  7. Oct 15, 2009 #6
    Sorry if i'm being stupid here;

    Could you please tell me how you got s=ln(cosh(t))? I evaluated the integral as C=ln(cosh(s)), so I am confused as to how you made this step. I'm okay as to how you solved the problem from there on though :)


    Thank you for the help,
    Oscar
     
  8. Oct 15, 2009 #7

    LCKurtz

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    Your original integral was (or should have been)

    [tex] s(t) = \int_0^t tanh(u)\, du [/tex]
     
  9. Oct 15, 2009 #8
    Ahh i see my error, you were right in your first post, I have misread the question.

    I understand now.

    Thank you for your help :)
    Oscar
     
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