Determine the length of the curve sin(x)

Loren Booda

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

disregardthat

It's [tex]\int^{2\pi}_0\sqrt{\cos(x)^2+1} dx[/tex]

Loren Booda

That's what I got. Would one need a table of integrals to determine its numerical value?

HallsofIvy

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

Yep. It's 4√2E(1/2), where E(x) is the complete elliptical integral of the second kind.

mathwonk

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.)