Short on time? This page is the recap. For the sector argument, the diagrams and the worked examples, use the full study guide. BC-only material. Prerequisites (polar coordinates 9.7, single-curve polar area 9.8, area between curves 8.4) are on the Calculus BC roadmap.
Recap
- A thin slice between two polar curves is a big sector minus a small sector: ½ R² dθ − ½ r² dθ.
- So the area between them is ½ ∫ from α to β of (R² − r²) dθ, with R ≥ r ≥ 0 on [α, β].
- Three ingredients every time: limits, outer curve, inner curve.
- A region inside both curves needs no subtraction. Split it at the intersection angle and use the curve nearer the pole on each piece.
Key relationships
| Situation | Set-up | Note |
|---|---|---|
| Inside f, outside g | ½ ∫ (f² − g²) dθ | Only over the angles where f ≥ g |
| Inside both f and g | ½ ∫ (nearer curve)² dθ, piece by piece | Split at each intersection angle |
| One curve only | ½ ∫ f² dθ | Topic 9.8 |
| Example: inside r = 2 + 2 cos θ, outside r = 3 | ½ ∫ from −π/3 to π/3 of [(2 + 2 cos θ)² − 9] dθ = 9√3/2 − π | Curves meet where cos θ = ½ |
Method
- Sketch both curves.
- Solve f(θ) = g(θ) for the intersection angles.
- Check the pole: where is each r equal to 0? Curves can share the pole at different angles.
- Test an angle in each interval to see which curve is outer (or nearer the pole).
- Write the integral(s), using symmetry if it is genuine.
- Evaluate by hand (standard angles) or by calculator (store the angles; 3 decimal places).
Mistakes to avoid
- ½ ∫ (R − r)² dθ. Square each radius first, then subtract.
- Forgetting the ½.
- Missing the pole as a shared point.
- Farther curve for “inside both”. Use the nearer one on each piece.
- 0 to 2π by habit. Some curves (r = a sin θ) are traced twice in a full turn.
- Rounded limits. Store the calculator’s angles; round only the final answer.
Quick self-check
- Find the area between r = 4 and r = 2. (½ ∫ from 0 to 2π of (16 − 4) dθ = 12π)
- Where do r = 2 + 2 sin θ and r = 3 meet? (sin θ = ½, so θ = π/6 and 5π/6)
- Find the area between r = 2θ and r = θ for 0 ≤ θ ≤ π. (½ ∫ from 0 to π of 3θ² dθ = π³/2, about 15.503)
Next: practice questions.