9.7
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…
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’s last known point acts as the pole for the polar curve. The drone’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.
View the full transcript and gain access to JoVE Core videos
Q1: How do polar coordinates describe a drone's position during a search mission?
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.
Q2: Why must polar equations be converted to rectangular coordinates to find arc length?
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.
Q3: What derivatives are needed when converting from polar to rectangular coordinates?
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.
Q4: How does the Pythagorean identity simplify the polar arc length formula?
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.
Q5: What does the polar arc length integral tell the rescue team about the drone mission?
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.
Q6: How does squaring and adding the rates of change produce the polar arc length formula?
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.
Q7: Why is the pole important when setting up a polar coordinate system for the search?
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.