Can someone please explain to me what rules have been used to calculate ${\bf{\dot{x}}}$ and $|{\bf{\dot{x}}}|$ in the definition for total arc-length of this problem. I've tried to calculate this by expanding everything out, but get lost in the workings. Has the identity $\cos^2t+\sin^2t=1$ been used to eliminate $\cos$ and $\sin$?
Example: Calculate the total length of the following curve.
The logarithmic spiral, $x=e^{−ωt}$ $(R\cos2πt, R\sin2πt)$ with ω > 0 and $R > 0$ constants and I = [0,∞).
Solution:
The total length is
\begin{align} L &=\int_0^∞ Re^{−ωt}(ω^2 + 4π^2)^{1/2} dt\\ &= \frac{R}{ω} (ω^2 + 4π^2)^{1/2}. \end{align}
$L = \int \|\frac {d\mathbf x}{dt}\| \ dt$
$\frac {d}{dt} \mathbf x = -\omega e^{-\omega t}(R\cos 2\pi t,R\sin 2\pi t) + e^{-\omega t}(-2\pi R\sin 2\pi t,2\pi R\cos 2\pi t)$
The two terms are orthogonal to one another.
$\|\frac {d\mathbf x}{dt}\|^2 = \|-\omega R e^{-\omega t}(\cos 2\pi t,\sin 2\pi t)\|^2 + \|2\pi Re^{-\omega t}(-\sin 2\pi t,\cos 2\pi t)\|^2$
$\|\frac {d\mathbf x}{dt}\| =Re^{-\omega t}\sqrt{\omega^2+(2\pi)^2} $