# Determine the length of the curve sin(x)

by Loren Booda
Tags: curve, determine, length, sinx
 P: 3,408 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: 3,408 That's what I got. Would one need a table of integrals to determine its numerical value?
Math
Emeritus
Thanks
PF Gold
P: 38,706

## 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: 14,243
 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.
 Sci Advisor HW Helper P: 9,406 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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