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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  
 =2cosdt  
 =2cosdt + 2 - cosdt  
 =22 sin +2 -2 sin  
 =12.