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

A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta

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.

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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).

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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.

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