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Math Help - Find area bounded by loop

  1. #1
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    Find area bounded by loop

    I can do the integral for these problems all day long but cannot determine the boundaries to save my life. Polar coordinates are my arch nemesis at the moment.

    The problem:
    find the area bounded by = r=3sin(3\theta)

    1. 1/2\int_a^b(3sin(3\theta))^2d\theta
    2. find boundaries
    3. do integral
    4. win

    I must have been catatonic when boundaries were explained because I simply do not get how to find them.

    thanks for any help!
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  2. #2
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    Re: Find area bounded by loop

    First, note that sin(\theta) goes from 0, up to +1, then down to 0 again as \theta goes from 0 to \pi. For \theta from \pi to 2\pi, sine is negative so you just go over the same path again. That is, sin(\theta) covers one loop as \theta goes from 0 to \pi. Here, \theta is replaced by 3\theta so this function covers one loop as 3\theta goes from 0 to \pi. That is, as \theta goes from 0/3 to \pi/3.
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  3. #3
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    Re: Find area bounded by loop

    I follow that. Does 3 at the beginning of the function not make a difference then?
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  4. #4
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    Re: Find area bounded by loop

    No, that does not affect the \theta values. In fact, it could be taken outside the integral.
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  5. #5
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    Re: Find area bounded by loop

    Oh, okay. So if I have a function like r=ksin(3\theta)cos(3\theta) I only need to look at sin(3\theta)cos(3\theta)? But this is only for multiplications of functions correct? In other words I could not pull out "n" or "k" in this example: r=n-ksin(3\theta)cos(3\theta). Do I have this right?

    And, in general form I could solve r=n-ksin(3\theta)cos(3\theta) by n=ksin(3\theta)cos(3\theta) and the solution to \theta would just be between 0 and 2\pi.

    Am I thinking of this correctly?
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