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

1. 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.

2. Originally Posted by eulcer
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

$z=3+2i$

$Re(z)=3 \ \ \ \ Im(z)=2$

3. This i is the same as the summation i? because it's reminding me of complex numbers.

4. Originally Posted by eulcer
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.

5. Ok, so how do imaginary or complex numbers come into play in a problem like this?

6. Originally Posted by Drexel28
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?
$\displaystyle \sum_{k=1}^n e^{\frac{i\pi k}{n}}\Rightarrow\sum_{k=1}^{n}\left[\cos{\left(\frac{\pi k}{n}\right)}+i\sin{\left(\frac{\pi k}{n}\right)}\right]$

7. 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?

8. Originally Posted by eulcer
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

9. 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.

10. Originally Posted by dwsmith
$\displaystyle e^{i\theta}=\cos{\theta}+i\sin{\theta}$

Exponential function - Wikipedia, the free encyclopedia
Was this your intended link?

Euler's formula - Wikipedia.

11. Originally Posted by TheCoffeeMachine
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.

12. 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.

13. Originally Posted by eulcer
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.

14. Originally Posted by dwsmith
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)?

15. Originally Posted by eulcer
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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