Polar Area (Two Curves)

Unit 9 · Parametric, Polar & Vector

Area Between Two Polar Curves

The area between an outer curve $r_{outer}$ and an inner curve $r_{inner}$ from θ = α to θ = β is A = (1/2)∫[α to β] ( $r_{outer}$ ² − $r_{inner}$ ²) dθ. Find the limits by solving $r_{outer} = r_{inner}$ .

Area Between Two Polar Curves

A = \frac{1}{2}\int_{\alpha}^{\beta}\left(r_{\text{outer}}^2 - r_{\text{inner}}^2\right)d\theta

AP Tip: Set the two r-expressions equal to find the angles where they intersect — these are your integration limits α and β.

Caution: Always verify which curve is outer (larger r) on the integration interval before subtracting.

Type 1

Setting Up the Area Integral

Find intersection angles, identify the outer and inner curves, then write the integral.

Example 1

Find the area inside r = 4 cos θ and outside r = 2.

Type 2

Using Symmetry to Simplify

When the region is symmetric about θ = 0 or θ = π/2, integrate over half the interval and double.

Example 2

Find the area inside r = 2 and outside r = 2 cos θ.

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