# Determine the length of the curve sin(x)

by Loren Booda
Tags: curve, determine, length, sinx
 P: 3,394 What is the measure of the sin(x) wave for x=0 to 2∏?
 Sci Advisor P: 1,670 It's $$\int^{2\pi}_0\sqrt{\cos(x)^2+1} dx$$
 P: 3,394 That's what I got. Would one need a table of integrals to determine its numerical value?
PF Patron
Thanks
Emeritus
P: 38,395

## Determine the length of the curve sin(x)

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
$$\int_{x=a}^b \sqrt{1+ f'(x)^2}dx$$
With y= f(x)= sin(x), f'(x)= cos(x) so that becomes
$$\int_{x=0}^{2\pi} \sqrt{1+ cos^2(x)}dx$$
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!
Mentor
P: 13,602
 Quote by HallsofIvy 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.
 HW Helper Sci Advisor P: 9,371 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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