Polar Area
Unit 9 · Parametric, Polar & Vector
Area in Polar Coordinates
The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = (1/2)∫[α to β] r² dθ. Find the limits by identifying where r = 0 or where the curve completes one loop. For symmetric curves, integrate over half and double.
Polar Area Formula
AP Tip: For petal curves r = a sin(nθ) or r = a cos(nθ): one petal spans an interval of length π/n. Set r = 0 to find the limits.
Caution: Don't forget the 1/2 factor — polar area is not simply ∫r dθ.
Type 1
Area of a Full Polar Curve
Integrate over the full period of the curve to find the total enclosed area.
Example 1
Find the area enclosed by r = 4 sin θ.
Type 2
Area of One Petal
Set r = 0 to find the limits of one petal, then integrate (1/2)∫r² dθ over that interval.
Example 2
Find the area of one petal of r = 3 sin(2θ).
Type 3
Area of a Cardioid
Cardioids r = a(1 ± sin θ) or r = a(1 ± cos θ) are traced once over [0, 2π]. Use the half-angle identity to evaluate.
Example 3
Find the area enclosed by r = 1 + cos θ.