Arc length
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Calculating arc length
Suppose we have a vector function \(\mathbf{r}:[a,b] \to \mathbb{R}^n\):
\[\mathbf{r}(t) = \left\langle r_1(t), r_2(t), \dots\right\rangle\]
where \(a \le t \le b\). We will assume that the components are differentiable.
We want to find the length of the curve from \(\mathbf{r}(a)\) to \(\mathbf{r}(b)\).
The idea is to first approximate the curve by a sequence of vectors like in this illustration:
We then calculate the sum of the magnitudes of all the vectors, and let the number of vectors go to infinity. More details and some examples in the following video: