The Archimedean spiral is given as $r=at,\, a>0, \, \text{for}\, t \in [0, \infty).$ I need to calculate the length of the first turn in the third quadrant. I have absolutely no idea how to compute that.
2026-03-29 20:03:14.1774814594
Archimedean spiral and length
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Hint:
The formula for arc length in polar coordinates is $$\ell=\int_\pi^{\tfrac{3\pi}2}\sqrt{r^2(t)+r'^2(t)}\,\mathrm dt.$$