Arc Length of Space Curves

Arc length represents the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be modeled by a vector-valued position function

\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}

where t denotes time. Unlike displacement, which measures only the straight-line distance between two points, arc length accounts for every change in direction along the trajectory.

To calculate arc length, the interval of motion is divided into many small time intervals of size Δt. Over each interval, the curve can be approximated by a short straight segment. The displacement during one interval is

\begin{equation*}\Delta \mathbf{r}_i=\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)\end{equation*}

Using the difference quotient,

\begin{equation*}\Delta \mathbf{r}_i=\left[\frac{\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)}{\Delta t}\right]\Delta t\end{equation*}

and for sufficiently small Δt,

\begin{equation*}\frac{\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)}{\Delta t}\approx\mathbf{r}'(t_i)\end{equation*}

Therefore, the length of each small segment can be approximated by

\begin{equation*}s_i \approx |\Delta \mathbf{r}_i|\approx|\mathbf{r}'(t_i)|\Delta t\end{equation*}

Adding all segment lengths produces a Riemann sum approximation for the total distance traveled:

\begin{equation*}S \approx \sum |\mathbf{r}'(t_i)|\Delta t\end{equation*}

Taking the limit as the number of intervals approaches infinity gives the exact arc length formula for a space curve:

\begin{equation*}S=\int_a^b |\mathbf{r}'(t)|\,dt\end{equation*}

If

\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}

then

\begin{equation*}|\mathbf{r}'(t)|=\sqrt{\left(\jfrac{dx}{dt}\right)^2+\left(\jfrac{dy}{dt}\right)^2+\left(\jfrac{dz}{dt}\right)^2}\end{equation*}

Thus, the arc length of a space curve from t=a to t=b is

\begin{equation*}S=\int_a^b\sqrt{\left(\jfrac{dx}{dt}\right)^2+\left(\jfrac{dy}{dt}\right)^2+\left(\jfrac{dz}{dt}\right)^2}\,dt\end{equation*}