1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Arc Length & Parametric Curves

  1. Apr 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the length of the curve y=x^2-4|x|-x from x=-4 to x=4.

    3. The attempt at a solution

    I realized there is a corner at x=0 so i tried to get around this by pluggin in x for x>=0 and -x for x<0. However, my integrals don't match the answer.

    http://texify.com/img/%5CLARGE%5C%21%5CLARGE%5C%21%5Cint_%7B-4%7D%5E%7B0%7D%20%5Csqrt%7B1%2B%282x-3%29%5E2%7D%20dx%20%2B%20%5Cint_0%5E%7B4%7D%20%5Csqrt%7B1%2B%282x-5%29%5E2%7D%20dx%20.gif [Broken]

    Which does not match the answer 19.56, what did I do wrong?


    Also, when I do try to find arc length in parametric curves and often more that not, a curve repeats itself in a given interval. How do I detect this and avoid it to just find length of one cycle?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 25, 2009 #2

    jhae2.718

    User Avatar
    Gold Member

    How is this curve parametric? x and y are not functions of t.

    Arc length for a curve on Cartesian coordinates is the integral from a to b of sqrt(1+(dy/dx)2) dx.

    How are x2-3x and x2-5x derivatives of y=x2-4|x|-x? You are finding the arc length of a different curve: x3/3-3/2x2+C on [-4,0) and x3/3-5/2x2+C on (0,4].

    If you were operating with parametric equations, Arc length for a curve is the integral from a to b of sqrt((dx/dt)2+(dy/dt)2) dt.
     
    Last edited: Apr 25, 2009
  4. Apr 25, 2009 #3
    my mistake I was so intent on writing it up that i forgot to take derivative. I did not do this on the actual work when trying to solve this problem.


    About hte parametric curves(this one isn't), when i'm given an x=f(t) and y=g(t) to find it in an interval say [a,b] where it repeats in the interval [a,b] say a few times, how will i know when to break it up to find one length of the cycle?


    sorry about the confusion
     
  5. Apr 25, 2009 #4

    jhae2.718

    User Avatar
    Gold Member

    No problem.

    I would look at the behavior of the curve and look for the point where it begins to repeat itself. I would try to find an approximation of where it begins to do so and then solve the equations for a value of t where it does so.

    Could you post an example of such a curve? That would help me in understanding exactly what you want, and it would give me something to work with to get you a better answer on what to do.

    If the case is a parametric function that moves to a value and then repeats along itself, find the endpoint. It may be a max min; try finding a value for which dy/dx=0.
     
  6. Apr 25, 2009 #5
    x=cos(3t)
    y=sin(3t)

    x interval [-1,1]
     
  7. Apr 25, 2009 #6

    jhae2.718

    User Avatar
    Gold Member

    To deal with the corner, we split y=x2-4|x|-x into a piecewise function:
    y=x2+4x-x=x2+3x on (-inf,0)
    and
    y=x2-4x-x=x2-5x on (0,inf)

    Remember, |x|=-x, -inf<x<0 and x, 0<x<inf

    You incorrectly removed the absolute value term. This should give you the correct answer when you take Sa0sqrt(1+(dy/dx)2)dx+S0bsqrt(1+(dy/dx)2)dx.

    Can't believe I missed that.
     
    Last edited by a moderator: May 4, 2017
  8. Apr 25, 2009 #7

    jhae2.718

    User Avatar
    Gold Member

    Looking at the graph of x=cos(3t) and y-sin(3t) on a t interval beginning at 0 and ending at some c such that the graph traces over itself, we notice that at t=0 the graph is located at (1,0) in Cartesian coordinates. Thus, the graph will repeat itself the next time x=1.

    Thus, we set x=cos(3t)=1. Solving for t, the first solution we get greater than 0 is the upper limit of integration. Then we take find the arc length from 0 to that value.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Arc Length & Parametric Curves
Loading...