Short on time? This page is the recap. For the sector diagram, explanations and worked examples, use the full study guide. BC-only material. Prerequisites (polar curves 9.7, Riemann sums and definite integrals 6.2–6.4) are on the Calculus BC roadmap.
Recap
- A polar region is bounded by rays from the pole and a curve r = f(θ), so slice it into thin sectors, not rectangles.
- A sector of radius r and angle Δθ has area ½ r² Δθ. Adding sectors gives a Riemann sum; its limit is the integral.
- The rate at which area is swept out is dA/dθ = ½ r².
- Limits matter more than the integration. Sketch first.
Key relationships
| Situation | Area | Note |
|---|---|---|
| Curve and two rays | ½ ∫ from α to β of r² dθ | Region swept once from α to β |
| Whole closed curve, r never 0 | ½ ∫ over one full trace | Often 0 to 2π, but check |
| One petal or loop | ½ ∫ between consecutive zeros of r | r = 0 at the pole |
| Symmetric region | 2 × (area of one half) | Only if the halves match |
| By hand | sin²kθ = ½(1 − cos 2kθ), cos²kθ = ½(1 + cos 2kθ) | Use before integrating |
Traced twice? The circle r = 6 cos θ is traced once on [0, π] (area 9π). On [0, 2π] the integral gives 18π, double the true area.
Mistakes to avoid
- Forgetting the ½ or forgetting to square r.
- Expanding (a + b cos θ)² wrongly: the middle term 2ab cos θ is easy to lose.
- 0 to 2π by habit, which double-counts curves that repeat.
- Wrong zeros for a petal: use consecutive zeros with the petal between them.
- Integrating sin²θ without the identity.
- Degree mode on the calculator, or giving a number with no integral written down.
Quick self-check
- Find the area of the region bounded by r = 5 and the rays θ = 0 and θ = π/2. (25π/4, a quarter disc)
- Find the area of one petal of r = sin 4θ. (Petal on 0 ≤ θ ≤ π/4: ½ ∫ sin²4θ dθ = π/16)
- Find the area swept out by r = e^θ for 0 ≤ θ ≤ 1. ((e² − 1)/4 ≈ 1.597)
- Why does ½ ∫ from 0 to 2π of (2 cos 3θ)² dθ not give the area of the rose? (The rose is traced twice on [0, 2π]; the integral gives 2π but the area is π.)
Next: practice questions.