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Math Help - How do I use Riemman sums to integrate (4-x^2)^(1/2) from 0 to 2?

  1. #16
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    I think this is a bit over my head, but I am well willing to try to make sense of it. I've seen the trig identity. I'm not real familiar with the proof but I remember making my way through it once before. I'm not familiar with the last bit in the slightest.
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  2. #17
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    Quote Originally Posted by eulcer View Post
    I think this is a bit over my head, but I am well willing to try to make sense of it. I've seen the trig identity. I'm not real familiar with the proof but I remember making my way through it once before. I'm not familiar with the last bit in the slightest.
    Here is simple example

    z=3+2i

    Re(z)=3 \ \ \ \ Im(z)=2
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  3. #18
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    This i is the same as the summation i? because it's reminding me of complex numbers.
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  4. #19
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    Quote Originally Posted by eulcer View Post
    This i is the same as the summation i? because it's reminding me of complex numbers.
    If you see in Drexel's post, there is a Re part.

    That is talking about the real part.

    This i in my example is about a complex number because with complex numbers we have a real part and an imaginary part.
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  5. #20
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    Ok, so how do imaginary or complex numbers come into play in a problem like this?
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  6. #21
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    Quote Originally Posted by Drexel28 View Post
    Note that \displaystyle \sum_{k=1}^{n}\cos^2\left(\frac{\pi k}{2n}\right)=\frac{1}{2}\sum_{k=1}^{n}\left(1+\co  s\left(\frac{\pi k}{n}\right)\right)=\frac{n}{2}+\frac{1}{2}\text{R  e}\sum_{k=1}^{n}e^{\frac{i\pi k}{n}}. Is any of that familiar looking to you?
    \displaystyle \sum_{k=1}^n e^{\frac{i\pi k}{n}}\Rightarrow\sum_{k=1}^{n}\left[\cos{\left(\frac{\pi k}{n}\right)}+i\sin{\left(\frac{\pi k}{n}\right)}\right]
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  7. #22
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    Is this Euler?
    I have a book by Dunham on him that I just picked up from the library but I haven't gotten far.
    How is this derived and where/when did you learn it?
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  8. #23
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    Quote Originally Posted by eulcer View Post
    Is this Euler?
    I have a book by Dunham on him that I just picked up from the library but I haven't gotten far.
    How is this derived and where/when did you learn it?
    \displaystyle e^{i\theta}=\cos{\theta}+i\sin{\theta}

    Exponential function - Wikipedia, the free encyclopedia
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  9. #24
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    Yeah. I was wondering how that was applied.
    I read the portion on logs, but I'm just starting complex numbers.
    I would never have guessed this problem would have come to this point.

    I read about the expansion of cosine and sin to infinite series today.
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  10. #25
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    Quote Originally Posted by dwsmith View Post
    Was this your intended link?

    Euler's formula - Wikipedia.
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  11. #26
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    Quote Originally Posted by TheCoffeeMachine View Post
    Was this your intended link?

    Euler's formula - Wikipedia.
    That works too but if you read over the exponential function it shows the identity too.
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  12. #27
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    Well, I suppose I can just read on and keep this in mind.
    My calc. teacher didn't know that e^i(pi) was equal to -1 so I don't think she'll be able to make much more out of this than myself.
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  13. #28
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    Quote Originally Posted by eulcer View Post
    Well, I suppose I can just read on and keep this in mind.
    My calc. teacher didn't know that e^i(pi) was equal to -1 so I don't think she'll be able to make much more out of this than myself.
    Maybe you should go to a different school.
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  14. #29
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    Quote Originally Posted by dwsmith View Post
    Maybe you should go to a different school.
    Next year I'll be done with HS, so I'll get a chance. When would you cover this material if you were majoring math (as I suspect most of you are or have)?
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  15. #30
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    Quote Originally Posted by eulcer View Post
    Next year I'll be done with HS, so I'll get a chance. When would you cover this material if you were majoring math (as I suspect most of you are or have)?
    Complex Numbers?
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