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Homework Help: Areas and Lengths in Polar Coordinates

  1. May 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the area of the region enclosed by one loop of the curve.
    r = sin(10θ)

    I can't seem to get the correct answer...I checked every step. I was not sure what to integrate from but the polar graph of sin(10θ) should be similar to polar graph of sin(2θ). From pi/2 to 0?

    2. Relevant equations
    A = integral(b to a) (1/2)r^2 dθ

    half angle formula
    (sinθ)^2 = (1/2)(1-cos2θ)dθ

    3. The attempt at a solution
    A = integral(b to a) (1/2)r^2 dθ = integral(pi/2 to 0) (1/2)r^2 dθ

    A = (1/2) integral(pi/2 to 0) (sin(10θ))^2 dθ
    A = (1/2) integral(pi/2 to 0) (1/2)(1-cos(20θ))dθ
    A = (1/4) integral(pi/2 to 0) (1-cos(20θ))dθ
    A = (1/4) [(θ-(1/20)sin(20θ)] (pi/2 to 0)
    A = (1/4) [(pi/2-(1/20)sin(20*(pi/2)) - (0 - 0)]
    A = (1/4)* (pi/2) = pi/8
     
    Last edited: May 11, 2010
  2. jcsd
  3. May 11, 2010 #2
    Nevermind. I figured it out. Integrating from pi/10 to 0. Thanks.
     
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