Applying the formula, we have for our Golden Spiral example:
From the question:
`r = 1.618013\ e^(0.30635\ θ)`
`r^2 = (1.618013\ e^(0.30635\ θ))^2` `= 2.61797\ e^(0.6127\ θ)`
`(dr)/(d theta)=0.49568\ e^[0.30635\ theta]`
`((dr)/(d theta))^2=0.245697\ e^[0.6127\ theta]`
Putting it together, the required length is:
`L` `=int_0^[4pi] sqrt[2.61797e^[0.6127theta]+0.2457e^[0.6127 theta]]d theta` `=254.0`
This is actually quite close to our very rough estimate before.
Note: The answer of 254.0 above comes from using a computer algebra system, like Scientific Notebook or Wolfram|Alpha (which gives quite a different answer, 244.2).
If you wanted to do this question by hand, you would use a numerical technique, like Simpson's Rule or Riemann Sums.
Easy to understand math videos: