I was wondering, if you have a simple Archimedean spiral, defined as : $ r=b\times \theta $ , with the angle $ \theta $ going from $ 0 $ to $ \Theta$
If you consider the spiral as a succession of infintesimally small circle arcs, couldn't you then just get the total length of the spiral by calculating the following integral :
$ \int_{0}^{\Theta}{r(\theta) \,d\theta} = \int_{0}^{\Theta}{b \times \theta \,d\theta} = \frac{1}{2}b \Theta ^2 $
But the Wikipedia page for Archimedean Spiral gives the following formula for the length of the spiral, derived from a cartesian parameterization of the spiral : $ {\displaystyle {\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\ln \left(\theta +{\sqrt {1+\theta ^{2}}}\right)\right]} $
Meaning my original formula is evidently false, and I suppose my error lies in considering the spiral as a succession of infintesimally small circle arcs.
Now, I don't really understand why exactly my reasoning is false.
Any help in understanding that would be much appreciated, thanks.
Treat the spiral as a succession of infinitesimal segments, taking into account that these segments are not perpendicular to the radius, so you should use the Pythagorean theorem (or some other apporach) to find their (infinitesimal) lengths. This way you'll get a more complicated but correct integral.