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Length of a Polar Curve

  1. Sep 8, 2010 #1
    1. The problem statement, all variables and given/known data
    r = 3 sin [tex]\vartheta[/tex]

    0 [tex]\leq[/tex] [tex]\vartheta[/tex] [tex]\leq[/tex] [tex]\pi[/tex]/3


    2. Relevant equations

    Arc Length: [tex]\int[/tex] [tex]\sqrt{r^{2} + (dr/d\vartheta)^{2}}d\vartheta[/tex]


    3. The attempt at a solution
    [tex]
    r^{2} = 9 (sin \vartheta)^{2} = 9 (1/2 - cos 2\vartheta/2)[/tex]

    [tex]r^{2} = 9/2 - 9/2 cos 2\vartheta[/tex]

    [tex]dr/d\vartheta = 3 cos \vartheta[/tex]

    [tex]\int (9/2 - 9/2 cos 2\vartheta + 3 cos \vartheta)^{1/2} d\vartheta[/tex]
    from 0 to \pi/3

    Not sure how to integrate this.
     
  2. jcsd
  3. Sep 8, 2010 #2

    Mark44

    Staff: Mentor

    You really made it hard on yourself.

    [tex]r^{2} = 9 sin^2( \theta)[/tex]
    [tex](dr/d\theta)^2 = (3 cos(\theta))^{2} = 9cos^2(\theta)[/tex]
     
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