Area in Polar Coordinates

AP Calculus · Axiom Academy

LESSON Area in Polar Coordinates Finding areas enclosed by polar curves To find the area enclosed by a polar curve, we divide the region into thin wedges. Each wedge has angle dθ and extends from the origin to the curve at distance r. A thin wedge is almost a sector of a circle. The area of a sector with angle dθ and radius r is approximately: To get the total area, we integrate these wedges from θ = a to θ = b: where a and b are the starting and ending angles This is one of the most elegant formulas in calculus! It naturally captures how area changes with angle in polar coordinates. The area of a sector is . Integrating gives the total area. Find the area enclosed by the circle . Perfect! We got the standard circle area formula. To find the area between two polar curves where r₂ > r₁: This is just the difference of two polar areas—the area of the outer region minus the area of the inner region.

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