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Find area of regions bounded by curve and line

  1. Dec 26, 2011 #1

    sharks

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    The problem statement, all variables and given/known data
    Find area of regions bounded by
    x^2 + y^2 = 9, y = 2x, x-axis in the first quadrant

    The attempt at a solution
    So, i drew the graph of y against x in my copybook, and circle with origin (0,0), radius = 3 units. The line y = 2x cuts through the circle.
    Transforming to polar coordinates, the new limits are:
    0≤θ≤tan^-1(3) and 0≤r≤3
    [tex]\int\int rdrd\theta[/tex]
    After integration, i get the final answer: (9/2)tan^-1(3)
    However, the answer in my notes is: (9/2)tan^-1(2). Did i copy the wrong answer or is my work wrong? I'm not sure.
     
  2. jcsd
  3. Dec 26, 2011 #2

    SammyS

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    Your answer is wrong, because the following is wrong
    ...the new limits are:
    0≤θ≤tan^-1(3)...​
     
  4. Dec 26, 2011 #3

    sharks

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    OK, so here is the graph:
    http://s2.ipicture.ru/uploads/20111227/zpUVSESp.jpg
    From my understanding, i need to find the area of the red section from the graph above.

    For θ fixed, 0≤θ≤cos^-1(1/√5) and 0≤r≤3
    I checked from calculator and cos^-1(1/√5) = tan^-1(2)

    Thanks for your help, SammyS.:smile:
     
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