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COV area under a curve

  1. Sep 14, 2010 #1
    So, If you've got two points and a given length of curve to 'hang' between them, what shape is the curve which minimises the area underneath it? For a curve which is almost the same length as the distance between the points, this would be a catenary, I think (a la famous hanging chain problem), but for longer curves it would be different. Any ideas?
     
  2. jcsd
  3. Sep 15, 2010 #2
    So you want to minimize the integral

    [tex]I=\int_a^b f(x)dx[/tex]

    with the constraints

    [tex]f(a)=A[/tex]

    [tex]f(b)=B[/tex]

    [tex]L=\int_a^b\sqrt{1+f'^2}dx[/tex]

    It's an Euler-Lagrange problem. The lagrangian is

    [tex]\mathscr{L}=f+\lambda\sqrt{1+f'^2}[/tex]

    so the equations of motion are

    [tex]\frac{d}{dx}\frac{\lambda f'}{\sqrt{1+f'^2}}=1[/tex]

    in other words

    [tex]\frac{\lambda f'}{\sqrt{1+f'^2}}=cx+d[/tex]

    You have to find c, d and lambda using the constraints above. Then you have to solve for f '. Finally you integrate (this is the hard part) to find f.
     
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