How long is a piece of sinosoidal string?

  • Thread starter Thread starter benjyk
  • Start date Start date
  • Tags Tags
    String
benjyk
Messages
10
Reaction score
0
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
 
Physics news on Phys.org
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)).
 
Right, there is no elementary answer
 
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.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top