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Parametric and Polar Curves

Problems

Length of a Parametric Curve

The Area Below a Polar Curve

Problem : Find the length of the parametric curve given by

,(2t + 1)3/2    

from t = 0 to 2 .

The length formula gives


dt = dt  
    = t + 1dt  
  = + x = 4  

Problem : Compute the length of (t + sin(t), 2 - cos(t)) from t = 0 to 3Π . You may use the trigonometric identity 1 + cos(θ) = 2 cos2(θ/2).

The velocity vector of this parametric curve at time t equals (1 + cos(t), sin(t)) , so the length of the curve is


dt =  
  = dt  
  = 2 cos dt  
  = 2cos dt + 2 - cos dt  
  = 22 sin +2 -2 sin  
  = 12.  

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