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Deriving the formula for arc length of a polar function

  1. Jan 23, 2015 #1
    1. The problem statement, all variables and given/known data
    Derive ∫(dr/dθ)^2 + R^2 )^0.5 dθ

    2. Relevant equations
    x = Rcosθ
    y = Rsinθ

    3. The attempt at a solution
    Arc length is the change in rise over run, which can be found using Pythagorean's Theorem. Rise is dy/dθ while run is dx/dθ. The arc length is [(dy/dθ)^2 + (dx/dθ)^2 ]^1/2

    dx/dθ = (cosθ -Rsinθ)
    dy/dθ = (sinθ + Rcosθ)
    dx/dθ ^2 + dy/dθ ^2 = (cos - Rsinθ)^2 + cos^2θ - 2Rsinθcosθ + R^2sin^2θ + sin^2θ + 2Rsinθcosθ + R^2cos^θ

    This simplifies to [R^2(cos^2θ + sin^2θ) + sin^2 θ+cos^2θ]^2/4
    Which leaves ∫√R^2 + 1 dθ
    But that is not the right formula!
     
  2. jcsd
  3. Jan 23, 2015 #2

    Mark44

    Staff: Mentor

    No, not at all. The arc length is the length along the curve between two points (r1, θ1) and (r2, θ2). You can approximate this length using the chord between these two points.
     
  4. Jan 23, 2015 #3
    This is certainly not correct. Could you check your differentiation?
     
  5. Jan 23, 2015 #4
    x = Rcosθ
    y = Rsinθ

    R is a constant, so by the multiplication rule of derivatives,
    dx/dθ = (1)cosθ + R(-sinθ) = cosθ - Rsinθ
    dy/dθ = (1)sinθ) + R(cosθ) = sinθ + Rcosθ

    I still get the same differentiation.
     
  6. Jan 23, 2015 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    R is not a constant, it's a function of theta. And even if it were, the derivative wouldn't be 1.
     
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