Determine the length of the curve sin(x)

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What is the measure of the sin(x) wave for x=0 to 2∏?
 

disregardthat

Science Advisor
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It's [tex]\int^{2\pi}_0\sqrt{\cos(x)^2+1} dx[/tex]
 
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That's what I got. Would one need a table of integrals to determine its numerical value?
 

HallsofIvy

Science Advisor
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Pretty straight forward, isn't it? Considering the other problems you have posted on here, you should be able to do this.

The length of the graph of y= f(x), from x= a to x= b, is given by
[tex]\int_{x=a}^b \sqrt{1+ f'(x)^2}dx[/tex]
With y= f(x)= sin(x), f'(x)= cos(x) so that becomes
[tex]\int_{x=0}^{2\pi} \sqrt{1+ cos^2(x)}dx[/tex]
However, that looks to me like a version of an elliptical integral which cannot be done in terms of elementary functions.

Hey, no fair posting while I'm typing!
 

D H

Staff Emeritus
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However, that looks to me like a version of an elliptical integral which cannot be done in terms of elementary functions.
Yep. It's 4√2E(1/2), where E(x) is the complete elliptical integral of the second kind.
 

mathwonk

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to get a numerical value try numerical integration, simpson's rule? etc...

this is no worse than finding the area under the curve from 0 to 1. i.e. both are approximations.


(nobody knows what cos(1) is.)
 

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