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Finding the length of a curve

  1. Jan 21, 2009 #1
    Question: Find the length of the curve 24xy = x^4 + 48 between (2, 4/3) and (3, 43/24). Answer is supposed to be 9/8.

    This is the formula (I think): square root(1 + (dy/dx)^2 dx)

    I tried to integrate and I did it like this:

    dy/dx = 3/24 x^2 - 2x^-2 or 3/24 x^2 - 2/x^2

    then: square root(1 + (3/24 x^2 -2x^-2)^2)

    = square root(1 + (1/64 x^4 - 4x^-4))

    now integration:

    (1 + (1/64 x^4 - 4x^-4))^(1/2)

    = (2/3) (1 + (1/64 x^4 - 4x^-4))^(3/2) (x + (1/320 x^5 + 4/3 x^-3)

    That's how far I got, I think it's wrong. If anyone could help me out, I would appreciate it. Thanks.
     
    Last edited: Jan 21, 2009
  2. jcsd
  3. Jan 21, 2009 #2
    How did you come up with your y'
     
  4. Jan 21, 2009 #3
    24xy = x^4 + 48

    y = (x^4 + 48) / 24x

    y = 1/24 x^3 + 2/x or 2x^-1

    dy/dx = 3/24 x^2 - 2x^-2
     
  5. Jan 21, 2009 #4
    Now show me how you do this:

    then: square root(1 + (3/24 x^2 -2x^2)^2)

    = square root(1 + (1/64 x^4 - 4x^-4))

    [You probably mean 1 + (3/24 x^2 - 2/x^2)^2)]
     
  6. Jan 21, 2009 #5
    the 1+ part isn't squared, just the inside bracket. I just left the square root part, but I only squared the inside.
     
    Last edited: Jan 21, 2009
  7. Jan 21, 2009 #6
    My post does not indicate that it is, does it?

    My first point was that you wrote 2x^2 whereas you meant to write 2/x^2 or 2x^(-2)

    My second point is show me how (3/24 x^2 - 2/x^2)^2 = 1/64 x^4 - 4x^(-4)
     
  8. Jan 21, 2009 #7
    is it: 1/64 x^4 - 1/2 + 4/x^4 ?
     
  9. Jan 21, 2009 #8
    That looks better
     
  10. Jan 21, 2009 #9
    ok so now I have:

    (2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)
     
  11. Jan 21, 2009 #10
    I have no idea how you got that, you have to show more steps, for example next step would be to show what happens when you add 1 to what you had and take sqrt
     
  12. Jan 21, 2009 #11
    We can't add the one, and we can't just take the square root either. Have to just integrate it straight from the square root and plug in the numbers to get the length.

    This is what I did for the integration

    (1/64 x^4 - 1/2 + 4/x^4)^(1/2)

    = (2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)
     
  13. Jan 21, 2009 #12
    ...why not?

    The steps are:

    1) Take the derivative
    2) Square it
    3) add 1 to it
    4) take square root of that
    5) integrate
     
  14. Jan 21, 2009 #13
    Ok here goes, I hope that works...

    (1 + (1/64 x^4 - 1/2 + 4/x^4)

    65/64 x^4 + 1/2 + 5/x^4

    so now square root of that:

    root(65/65) x^2 + root (1/2) + root (5) / x^2


    that looks bad, that's why I thought to do it the other way:

    (1/64 x^4 - 1/2 + 4/x^4)^(1/2)

    integrates to:

    (2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)
     
  15. Jan 21, 2009 #14
    You should know from algeba that sqrt(a+b) is NOT sqrt(a) + sqrt(b), stop thinking that.

    To work with what you have, find the common denominator and then use the property that sqrt(a/b) DOES equal sqrt(a)/sqrt(b)
     
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