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Homework Help: Polar Coordinates - Areas

  1. May 7, 2012 #1
    I am asked to consider the following graph:

    r2=a+sin(θ), where a=2

    I have a picture of this plot, which I have attached,

    We are asked to find the area of the upper 'cresent' of the curve, contained at the top

    How would I go about calculating that?

    I've found that if I plot r=√(2+sinθ) and r=-√(2+sinθ) that this gives me the separate graphs individually, and that if I integrate from 0-π on the first one, then integrate from π-2π on the second one, if i subtract, i get the correct answer geometrically, using my graphing programme, but i dont know how to do this analytically? (i have attached another image to show the to sections - its the blue section i need)

    Any help would be vastly appreciated, thank you

    Attached Files:

  2. jcsd
  3. May 7, 2012 #2
    From my graphing programme, ive found the area should be 4.06-2.139 = 1.921 approximately, if this helps anyone?
  4. May 7, 2012 #3
    I think your going to have to use cylindrical coordinates. From the graph it's evident that the limits of theta would be 0 to pi. The limits of r would be sqrt(2) to sqrt(3) (if your confused about how did I get these limits try finding the max and min values of r by playing around with the theta value).
  5. May 7, 2012 #4
    I get where you have your limits by maximising sin in the range of 0-pi, giving r^2= 2 or 3... Ive never done cylindrical coordinates before? We've never encountered them in lessons yet. Worrying
  6. May 7, 2012 #5
    Never mind then lol. I know its definitely not cylindrical coordinates. Though.
  7. May 10, 2012 #6
    Is there any reason you are plotting [itex]r = -\sqrt{2+sinθ}[/itex]? It would be easier if you use [itex]r = \sqrt{2-sinθ}[/itex]. It plots the same circle but the intersecting points are coincident. The you could simply integrate


    Since these two curves have the same period and starting point you can integrate from 0 to ∏
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