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Homework Help: Area in polar (stuck at the intersection points!)

  1. May 30, 2010 #1
    1. The problem statement, all variables and given/known data
    find the area inside both of the curves
    r = 4 cos@
    r = 2+2cos@
    @ = theta

    2. Relevant equations

    3. The attempt at a solution
    i will say 4cos@ = 2+2cos@ to find the intersection points
    4cos@ = 2+2cos@
    2cos@ = 2
    cos@ = 1
    @ = 0 !!
    I need the other points!!
  2. jcsd
  3. May 30, 2010 #2


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    Yes θ=0 is one point. Now just add π/2 to your principal answer of 0 and that will give you another answer.
  4. May 30, 2010 #3


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    Gold Member

    You have to be careful with this type of problem. Eyes can be deceiving. Have you drawn a plot of the two curves? (I will use t for the angle). If so you will have noted that at t = 0 the r for both equations is 4, which is the point you have found. And the graphs both touch at the origin. The trouble is, at the origin t can be anything. And these two graphs do not have r = 0 for the same value of t. That is why you are having trouble finding the other point. What this means is you can't find the area with a single integral of the form

    [tex]\frac 1 2 \int_{\alpha}^\beta r_{outer}^2-r_{inner}^2\, dt[/tex]

    So do the areas separately, each with their correct limits and subtract the inner area from the outer one. You might notice as t goes from 0 to [itex]2\pi[/itex], one of the curves is traced twice.
  5. May 31, 2010 #4
    why Pi/2 and not 2Pi? I would rather have said that cos θ has two easy roots: 0 and 2Pi, and all multiples of 2Pi.
  6. May 31, 2010 #5


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    Sorry, I was solving cos(t)=0 not 1. You are right it would be 0 and 2π
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