Integration Applied to Polar Coordinates to Find Arc Lengths

In polar coordinates, a plane curve is described by a radial distance r from a fixed point, called the pole, and an angle θ measured from a reference direction. This system is especially useful for paths that naturally involve rotation, such as an expanding spiral followed by a search drone. If the hiker’s last known position is treated as the pole, then the drone’s location at any instant can be represented by the polar equation r = f(θ), where the distance from the pole changes as the drone rotates counterclockwise.

To compute the total distance traveled, the polar curve can be connected to rectangular coordinates using

\begin{equation*}x = r\cos\theta\end{equation*}

\begin{equation*}y = r\sin\theta\end{equation*}

Since r depends on θ, both x and y also change with θ. Differentiating gives

\begin{equation*}\jfrac{dx}{d\theta} = \jfrac{dr}{d\theta}\cos\theta - r\sin\theta\end{equation*}

\begin{equation*}\jfrac{dy}{d\theta} = \jfrac{dr}{d\theta}\sin\theta + r\cos\theta\end{equation*}

These expressions describe the rates of change of the drone’s horizontal and vertical positions as the angle changes.

The rectangular arc length formula is

\begin{equation*}ds = \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}\, d\theta\end{equation*}

Substituting the derivatives and simplifying produces

\begin{equation*}S = \int_{{\theta}_{1}}^{{\theta}_{2}} \sqrt{r^2 + \left(\jfrac{dr}{d\theta}\right)^2} \, d\theta\end{equation*}

The simplification occurs because the mixed trigonometric terms cancel, and the identity

\begin{equation*}\sin^2\theta + \cos^2\theta = 1\end{equation*}

combines the remaining terms. For the drone’s spiral route, evaluating this integral from the initial angle to the final angle gives the total flight distance, allowing the rescue team to estimate whether the battery capacity is sufficient for the mission.