9.9 Finding the Area of the Region Bounded by Two Polar Curves

Setting Up the Region Between Two Curves

To find area between two polar curves, first find their intersection angles by setting r₁(θ) = r₂(θ) and solving for θ. The area between the outer curve r_outer and inner curve r_inner from θ = a to θ = b is A = (1/2)∫[a,b] (r_outer² − r_inner²) dθ. Always identify which curve is farther from the origin on the interval before subtracting.

Watch for Symmetry and Multiple Regions

Many polar area problems involve symmetry (about the x-axis, y-axis, or origin), which lets you compute one region and multiply. Some regions require splitting the integral into pieces if the outer/inner roles switch, or if the curves overlap in petals or loops. Sketching the curves is essential to avoid subtracting the wrong function or missing an interval.

Common Pitfalls

Be careful with negative r values and multiple intersection points, since equations like r₁ = r₂ may have extra solutions from periodicity. Always check the graph to confirm which curve bounds the region and the correct θ-interval. Units are in radians, and integrals often require trig identities (like r² = (something)² using half-angle formulas) to evaluate.

Key terms

Area between two polar curves formula

A = (1/2)∫[a,b] (r_outer² − r_inner²) dθ, where a and b are the intersection angles.

How to find bounding angles

Set r₁(θ) = r₂(θ) and solve for θ, then verify solutions graphically.

Why sketch the curves first?

To correctly determine which curve is outer vs. inner and to avoid missing regions or intervals where roles switch.

Using symmetry in polar area problems

If the region is symmetric, compute the area of one symmetric portion and multiply by the number of symmetric parts.

Common pitfall with r₁ = r₂ solutions

Trig periodicity can create extraneous solutions; always confirm with the graph which intersections bound the actual region.

Try an AP-style question

Which formula gives the area between an outer polar curve R and an inner curve r?

1. π∫ (R² − r²) dθ
2. (1/2)∫ (R² − r²) dθ
3. ∫ (R − r) dθ
4. (1/2)∫ (R − r)² dθ