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_outer² − r_inner²) dθ. Find the limits by solving r_outer = r_inner.
Area Between Two Polar Curves
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.
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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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