BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (Topics 8.4, 9.7 and 9.8) are on the Calculus BC roadmap.
Understanding
- I can explain, using a big sector minus a small sector, why the area between two polar curves is ½ ∫ (R² − r²) dθ. (Guide: “From one curve to two”)
- I can explain why ½ ∫ (R − r)² dθ is wrong, and recognise the answer it produces. (Guide: “From one curve to two”; Practice Q1, Q4)
- I can explain why a region inside both curves needs splitting, not subtraction. (Guide: “Step 3: identify the type of region”; Practice Q3)
Skills
- I can find the angles where two polar curves meet by solving r₁(θ) = r₂(θ). (Guide: “Step 1”; Practice Q5(a), Q6(a))
- I can check whether two curves also share the pole, and count the distinct shared points. (Guide: “Step 1”; Practice Q2)
- I can set up and evaluate exactly the area inside one curve and outside another, using a double-angle identity. (Guide: Worked example 1; Practice Q1, Q5)
- I can find the area inside both curves by splitting at the intersection angle and using the nearer curve on each piece. (Guide: Worked example 2; Practice Q3, Q6)
- I can use symmetry correctly, stating which line the region is symmetric about. (Guide: “Using symmetry and the calculator well”; Practice Q5(d), Q6(c))
- I can use a graphing calculator to find intersection angles and polar areas, storing the angles and giving 3 decimal places. (Guide: Worked example 2; Practice Q4, Q7)
Reasoning
- I can justify which curve is outer (or nearer the pole) on an interval with a test angle. (Guide: “Step 2”; Practice Q6(b), Q7(b))
- I can choose limits so that each part of a region is swept exactly once, including when r would be negative. (Guide: “A note on negative r”; Practice Q5(d))
- I can use polar areas to answer a question in context, with units and a clear conclusion. (Practice Q7)
All ticked? Unit 9 is complete. Move on to Unit 10 with Topic 10.1, Defining Convergent and Divergent Infinite Series, or return to the Calculus BC roadmap.