Integration Applied to Polar Coordinates to Find Arc Lengths

Polar coordinates describe the drone's position using two values: the radial distance r from a fixed point called the pole, and an angle θ measured counterclockwise from the positive x-axis. For a search drone following an expanding spiral path, the hiker's last known location serves as the pole. This system naturally represents rotating paths where the distance from the pole changes as the drone rotates.

The arc length formula is derived from rectangular coordinates using derivatives of x and y with respect to a parameter. Since polar equations express r as a function of θ, converting to rectangular coordinates using x = r cos θ and y = r sin θ allows us to apply the standard arc length formula. This conversion bridges the two coordinate systems to calculate total distance traveled.

When r depends on θ, the derivatives are 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's horizontal and vertical positions change as the angle increases. Computing these rates of change is essential for applying the arc length formula in polar form.

After substituting the derivatives into the arc length formula and squaring them, trigonometric terms appear that can be simplified using sin²θ + cos²θ = 1. This identity causes mixed trigonometric terms to cancel and combines remaining terms, yielding the streamlined polar arc length formula: S = ∫√(r² + (dr/dθ)²) dθ. This simplification makes the integral computationally manageable.

Evaluating the integral S = ∫√(r² + (dr/dθ)²) dθ from the initial to final angle gives the total distance the drone must travel along its spiral route. The rescue team uses this distance to estimate flight time and determine whether the battery capacity is sufficient to complete the search mission without running out of power.

Squaring dx/dθ and dy/dθ, then adding them together creates an expression with trigonometric terms. When this sum is placed under a square root in the arc length formula, the Pythagorean identity simplifies the complex trigonometric expression. The result is the elegant polar formula that depends only on r and dr/dθ, eliminating explicit trigonometric functions.

The pole serves as the reference point from which all distances are measured. By placing the pole at the hiker's last known location, the rescue team can describe the drone's expanding spiral path relative to that critical point. This choice of pole makes the polar equation r = f(θ) naturally represent the drone's search pattern and simplifies distance calculations.