How long is a piece of sinosoidal string?

[benjyk]

I am trying to calculate the arc length of a sine wave.

Using $$s=\int_{}^{}\sqrt[]{1 + {(\frac{dy}{dx})}^{2}}dx$$

if y = sinx, dy/dx = cosx

So the integral simplyfies to $$s=\int_{}^{}\sqrt[]{1 + {cos}^{2}(x)}dx$$

However I do not know any integration technique (ie. substitution, by parts, etc..) with which I can calculate this integral analytically.

If you can think of any other way of going about this, any help would be greatly appreciated.

Benjy

 

[CompuChip]

Mathematica calls this function EllipticE (performing the integral from 0 to 2п gives $4\sqrt{2}\mathtt{EllipticE}(1/2) \approx 7.6404$), so I doubt there is a more elementary answer (like $\pi/2$ or $\operatorname{arcsinh}(-1)$).

 

[Midy1420]

Right, there is no elementary answer

 

[benjyk]

I have to admit I am a little disappointed. I thought there might be a way of performing the integral by pure analytical means.
But thank you very much for your responses.