# 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 $$\int^{2\pi}_0\sqrt{\cos(x)^2+1} dx$$

#### Loren Booda

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

#### HallsofIvy

Homework Helper
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!

#### D H

Staff Emeritus
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.