Results 1 to 3 of 3

Math Help - Help with arc length of parametric curves

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    7

    Help with arc length of parametric curves

    x(t) = 2t - 2sin(t)
    y(t) = 2 - 2cos(t)

    t is from 0 to 2Pi

    Okay, I am able to set up the problem correctly:

    x'(t) = 2 - 2cos(t)
    y'(t) = 2sin(t)

    Length = Integral from 0 to 2Pi of (Sqrt[[2 - 2cos(t)]^2 + [2sin(t)]^2])
    = Integral from 0 to 2Pi of (Sqrt[4[1-cos(t)]^2 + [2sin(t)]^2])

    I know I can pull the 4 out (making it a 2), but I'm stuck after that. What good does making a substitution of 1 - cos(t) = 2sin(t/2)^2 do?

    Help would be greatly appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Dec 2008
    Posts
    111
    Dear Drumiester,

    next step: bring out 4 from root
    next step: break down the bracket under the root and simplify
    next step: you can bring out again 2 from root
    you get \sqrt{1-cos(t)}

    Now one trick: cos(t) = cos(2*t/2) = cos^2(t/2) - sin^2(t/2)
    and
    1 = cos^2(t/2) + sin^2(t/2)
    So
    \sqrt{1-cos(t)} = \sqrt{cos^2(t/2)+sin^2(t/2) - cos^2(t/2) + sin^2(t/2)}  = \sqrt{2}*sin(t/2)

    Now it's not so wild, isn't it?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2008
    Posts
    7
    Thanks man! I just discovered this site, so I'll be sure to come here for help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Parametric curves and arc length...?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 27th 2010, 12:22 PM
  2. Parametric curves
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 14th 2010, 02:56 PM
  3. Parametric Curves
    Posted in the Calculus Forum
    Replies: 9
    Last Post: October 20th 2009, 07:14 PM
  4. Parametric Curves
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 6th 2009, 11:34 PM
  5. parametric curves
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 6th 2008, 11:35 PM

Search Tags


/mathhelpforum @mathhelpforum