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Math Help - Teacher needs help with polar co-ordinates

  1. #1
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    Teacher needs help with polar co-ordinates

    Haha. I bet this will be one of the more unusual questions you get on here. I am a trainee teacher and I noticed last week that one of the other teachers was asked a question about polar co-ordinates. He couldn't do it, and after showing it to the rest of us in the staffroom, we decided we couldn't either! Any help or guidance you could give would be very appreciated. By next week the student will probably be on the next chapter, but for the sake of our sanity we would like to see how to do it!

    The question is:

    Integrate (a^2) / (1 + cos theta) between 0 and pi/2 where a is just a constant.

    Now we know that an integration in polar co-ordinates is

    A = (a^2) / 2 x integral 1 / (1+cos theta)^2 d-theta

    Everything we try from there onwards just makes things much worse. Substitution doesn't help, so I was thinking there could be some trig-identity which would make things easier but I couldn't see one.

    Thanks loads.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by TheBrain View Post
    Haha. I bet this will be one of the more unusual questions you get on here. I am a trainee teacher and I noticed last week that one of the other teachers was asked a question about polar co-ordinates. He couldn't do it, and after showing it to the rest of us in the staffroom, we decided we couldn't either! Any help or guidance you could give would be very appreciated. By next week the student will probably be on the next chapter, but for the sake of our sanity we would like to see how to do it!

    The question is:

    Integrate (a^2) / (1 + cos theta) between 0 and pi/2 where a is just a constant.

    Now we know that an integration in polar co-ordinates is

    A = (a^2) / 2 x integral 1 / (1+cos theta)^2 d-theta

    Everything we try from there onwards just makes things much worse. Substitution doesn't help, so I was thinking there could be some trig-identity which would make things easier but I couldn't see one.

    Thanks loads.
    sir,
    \int_0^{\frac{\pi}{2}} \frac {a^2}{1+cos\theta}d\theta

    a^2 is a constant hence it can go out in the integral sign.. if you multiply it by \frac{1-cos\theta}{1-cos\theta} i believe you'll get a result..
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  3. #3
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    Sorry kaalgota maybe I wasn't clear enough or I am still confused. I thought that when you integrate something in polar co-ordinates you have to square the function and divide it by 2. That means you have to integrate something squarey on the bottom.
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  4. #4
    MHF Contributor kalagota's Avatar
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    pardon my young mind, but, aren't they the same? i am now confused.. or, maybe i have to check my notes..
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  5. #5
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    <br />
1/2 \int_0^{\frac{\pi}{2}} \frac {a^2}{(1+cos\theta)^2}d\theta<br />

    <learns how to use the formula thing>

    Well this is the integral that I end up with.
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  6. #6
    MHF Contributor kalagota's Avatar
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    if the question was to find the area of that function r=f(\theta) = \frac{a^2}{1+cos\theta} on the given interval, then it must be correct.. however, if you set-up the integral as it is, then i think, you'll just integrate it as it is..
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  7. #7
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    The actual question was

    <br />
r=f(\theta) = \frac{a}{1+cos\theta}<br />

    integrated between the limits.
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  8. #8
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by TheBrain View Post
    The actual question was

    <br />
r=f(\theta) = \frac{a}{1+cos\theta}<br />

    integrated between the limits.
    in my opinion, i'll just need to integrate it as it is (and not abuse the formula) since it didn't asked me to find the area of that curve from 0 to \frac{\pi}{2}.
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by TheBrain View Post
    <br />
1/2 \int_0^{\frac{\pi}{2}} \frac {a^2}{(1+cos\theta)^2}d\theta<br />

    <learns how to use the formula thing>

    Well this is the integral that I end up with.
    Why do you think this is anything to do with polars, \theta in:

    "Integrate (a^2) / (1 + cos theta) between 0 and pi/2 where a is just a constant."

    is just a dummy variable of integration. The integral you want is:

    \int \frac{a^2}{1+\cos(\theta)}~d\theta

    Nothing more.

    RonL
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  10. #10
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    I built the integral from the question and it is definately polar co-ordinates because it is from that chapter of the book and the integral limits are stated as being between angles pi/2 and 0. The integral that needs to be solved is definately

    <br />
1/2 \int_0^{\frac{\pi}{2}} \frac {a^2}{(1+cos\theta)^2}d\theta<br />
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  11. #11
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    \int {\frac{1}{{\left( {1 + \cos (x)} \right)^2 }} = \frac{1}{2}\tan \left( {\frac{x}{2}} \right) + \frac{1}{6}\tan ^3 \left( {\frac{x}{2}} \right)} <br />

    Now work backwards to see how the antiderivative works.
    BTW. Buy youcself a computer algebra system. There is one for @$40 if you have a PC.
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  12. #12
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    You can let {\theta}=2tan^{-1}(u), \;\ d{\theta}=\frac{2}{u^{2}+1}du, \;\ u=tan(\frac{\theta}{2})

    After making the subs, we get:

    \frac{a^{2}}{2}\int{\left[\frac{1}{2}u^{2}+\frac{1}{2}\right]}du

    \frac{a^{2}}{2}\left[\frac{1}{6}u^{3}+\frac{1}{2}u\right]

    Now, resub and get:

    \frac{a^{2}}{2}\left[\frac{1}{6}tan^{3}(\frac{\theta}{2})+\frac{1}{2}ta  n(\frac{\theta}{2})\right]

    Now, use the limits of integration and it boils down to \frac{a^{2}}{3}
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  13. #13
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by TheBrain View Post
    I built the integral from the question and it is definately polar co-ordinates because it is from that chapter of the book and the integral limits are stated as being between angles pi/2 and 0. The integral that needs to be solved is definately

    <br />
1/2 \int_0^{\frac{\pi}{2}} \frac {a^2}{(1+cos\theta)^2}d\theta<br />

    will you qoute the whole question..
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