Arc Length and the business of backtracking Provided some parameterized curve in space along which a particle travels in time, such as $\vec{r}(t) = (x(t), y(t))$.  Does the arc length formula take into account the fact that the particle may or may not backtrack.
i.e. does $\int_{0}^{t} |\vec{r}\prime(t)| ds$ give the length of the curve or the distance the particle traveled, which may be longer than the length of the curve (if the particle is able to backtrack).
It seems to me since $|\vec{r}\prime (t)|$ is the speed of the particle at time t, that this would be the distance the particle traveled including the backtracking.  How then, would I obtain a pure length of the curve itself?
 A: Here's a simple example showing that the integral gives the total distance traveled. Consider the line integral along the line segment in the real plane from the origin to $(1,1)$. From $(0,0)$ to $(1,1)$, you have $\vec{r}(t)=\langle t,t\rangle$ with $t\in[0,1]$. In the reverse direction, you would have $\vec{r}(t)=\langle 1-t,1-t\rangle$ with $t\in[0,1]$ and the particle starts at $(1,1)$ and travels to $(0,0)$. I'll denote the first path by $\mathcal{C}$ and its reverse by $-\mathcal{C}$.
Compute the line integral:
$$\int\limits_\mathcal{C}\,\mathrm{d}\vec{r}=\int_0^1\sqrt{1^2+1^2}\,\mathrm{d}t=\sqrt2$$
And in the reverse direction,
$$\int\limits_\mathcal{-C}\,\mathrm{d}\vec{r}=\int_0^1\sqrt{(-1)^2+(-1)^2}\,\mathrm{d}t=\sqrt2$$
Now consider both paths taken at the same time, so the particle starts at the origin, travels to $(1,1)$, then backtracks and returns to the origin. This compound path (which I denote by $\mathcal{C}+(-\mathcal{C})$) can be parameterized by $\vec{r}(t)=\left\langle 1-2\left|\frac{1}{2}-t\right|,1-2\left|\frac{1}{2}-t\right|\right\rangle$ with $t\in[0,1]$. The distance traveled is then
$$\int\limits_{\mathcal{C}+(-\mathcal{C})}\,\mathrm{d}\vec{r}=\int_0^{1/2}\sqrt{2^2+2^2}\,\mathrm{d}t+\int_{1/2}^1\sqrt{(-2)^2+(-2)^2}\,\mathrm{d}t=2\sqrt2$$
whereas you would intuitively expect the net distance traveled to be $0$. The notion of "arc length" would have to be carefully defined - does it refer to the net or total distance?
To summarize: If the particle does not change direction, then the net and total distances are the same. If it does change direction, then the net and total are different, and you need to specify which of these is equivalent to the length of the path.
