Name: polar curve distance with calculus
Uploaded: 2026-06-08T09:38:49.530Z
Duration: 1 min 26 s
Channel: JoVE

Consider a drone launched to find a lost hiker in a dense forest. To ensure every square meter of ground is inspected, the drone follows an expanding spiral path.Here, the hiker&#x2019;s last known point acts as the pole for the polar curve. The drone&#x2019;s position is defined by its distance from that point and the angle of its rotation measured counterclockwise from the positive x-axis.To determine if the drone has enough battery to finish the mission, the rescue team must calculate the total distance to be travelled using the arc length function.The derivation begins by using the relationship between polar and rectangular coordinates to find the rates of change for the horizontal and vertical positions.Squaring these rates and adding them together allows the formula to be expressed entirely in terms of polar coordinates.Here, the Pythagorean identity is used to simplify the expression. This identity collapses the complex trigonometric terms, leaving a streamlined formula for the total length of the curve.By solving this integral over the interval from the initial to the final angle, the team knows exactly how much flight time is required for the drone to complete the flight.

integration applied to polar coordinates to find arc lengths

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 &#x3B8; 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&#x2019;s last known position is treated as the pole, then the drone&#x2019;s location at any instant can be represented by the polar equation r = f(&#x3B8;), 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 &#x3B8;, both x and y also change with &#x3B8;. 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&#x2019;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&#x2019;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.

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integration-applied-to-polar-coordinates-to-find-arc-lengths

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JoVE Core

Calculus

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Parametric Equations and Polar Coordinates

parametric-equations-and-polar-coordinates

KEY TERMS AND CONCEPTS

polar curve distance with calculus

Polar coordinates can be used to measure the distance along a curve that turns around a fixed point. In this system, a point is described by the radial distance r from the pole and the angle θ from a reference direction. That makes it useful for motion such as an expanding spiral, where distance and direction change together.

A search drone gives a clear example of a polar path. If the hiker’s last known position is the pole, the drone’s location at any moment can be written as the polar equation r = f(θ). As the drone moves counterclockwise, its distance from the pole changes with the angle.

To find the total distance traveled, the polar curve is linked to rectangular coordinates. The relationships x = r cos θ and y = r sin θ show how the position in the plane depends on θ. Because r changes with θ, both x and y also change as the angle changes.

Differentiating gives the rates of change of the horizontal and vertical positions. The result is dx/dθ = (dr/dθ) cos θ - r sin θ and dy/dθ = (dr/dθ) sin θ + r cos θ. These expressions describe how the drone moves in each direction as it rotates.

The rectangular arc length formula is then used. It begins with ds = \sqrt{(dx/dθ)^2 + (dy/dθ)^2}\, dθ. After substituting the derivatives and simplifying, the polar arc length formula becomes S = \int_{θ_1}^{θ_2} \sqrt{r^2 + (dr/dθ)^2} \, dθ.

The simplification works because the mixed trigonometric terms cancel. The identity sin^2θ + cos^2θ = 1 also combines the remaining terms. Evaluating the integral from the initial angle to the final angle gives the total flight distance, which can help a rescue team estimate whether the drone has enough battery for the mission.

Polar Curve Distance with Calculus

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key terms and concepts

Learn how polar coordinates and calculus work together to find the distance along a spiral path using the arc length integral and derivative of r.

Polar Curve Distance with Calculus | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

Polar Curve Distance with Calculus

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