Area Inside Polar Curves

Trigonometry · Axiom Academy

LESSON Area Inside Polar Curves Discovering how to find areas in polar coordinates using the sector addition method Given a polar curve r = f(θ) , we want to find the area enclosed between angles θ = α and θ = β . Let's divide the angle interval [α, β] into n equal parts , each with width Δθ = (β - α)/n . At each angle θᵢ, we create a circular sector with radius r = f(θᵢ) . Key fact: The area of a circular sector with radius r and angle Δθ (in radians) is ½r²Δθ . As we make the sectors thinner and thinner (n → ∞), our approximation becomes exact. Watch how infinitesimal sectors perfectly fill the region: The total area is the sum of all sector areas : Taking the limit as n → ∞ (or equivalently, as Δθ → 0), we get the beautiful integral formula for area in polar coordinates: Why r² and not just r? Because we're accumulating areas of sectors, not just lengths. A sector with twice the radius has four times the area.

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