I want to calculate the arc-length of the Log spiral given by the equation:
$\vec{x}(t)=\begin{pmatrix} e^{-\alpha t} \cos t \\ e^{-\alpha t} \sin t\end{pmatrix}$
$\alpha >0$ and $t \in [0,T]$
Our lecturer gave us the following formula to work with:
$\int_{t_1}^{t_2}f(x)||\dot{x}(t)||dt$
However, in this case I don't really have a traditional function to work with. How does this work with vectors?
I am also not sure how to get to the formula. All I know is that:
1) I have some path/image of a function on the intervall [a,b] and I want to find the length of it
2) I can apporximate the length of the path by using small polygonal chains.
$\implies L(x)=\sum ||x(t_a)-x(t_{a-1})||$
But how do I go from that to the integral formula?
Best Answer
it should be easier to do in polar coordinates. the curve $$e^{-at}(\cos t, \sin t) \to r = e^{-a\theta}$$ the arc length formula is $$ ds = \sqrt{(r d \theta)^2 + (dr)^2} =\sqrt{a^2e^{-2a\theta}+e^{-2a\theta}}\, d\theta$$ therefore the arc length is $$\int_{t_1}^{t_2} e^{-a\theta}\sqrt{a^2+1}\, d\theta = \frac1a\sqrt{a^2 + 1}\left(e^{-at_1} - e^{-at_2}\right)$$