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 (Topic 9.7 and Riemann sums, Topics 6.2–6.4) are on the Calculus BC roadmap.
Understanding
- I can explain why polar regions are sliced into sectors rather than rectangles, and where ½ r² Δθ comes from. (Guide: “Why rectangles do not fit” and “Building the formula from thin sectors”; Practice Q1)
- I can explain the polar area formula as the limit of a Riemann sum of sectors. (Guide: “Building the formula from thin sectors”, Figure 1)
- I can explain why ½ r² is the rate at which area is swept out as θ increases. (Guide: “Area as an accumulation”; Worked example 2(b))
Skills
- I can find the area between a polar curve and two rays, expanding r² and using the power-reducing identities. (Guide: Worked example 1; Practice Q5)
- I can find the area of one petal of a rose by using consecutive zeros of r as limits. (Guide: “Choosing the limits”; Practice Q3)
- I can find the area enclosed by a circle or closed curve by choosing an interval that traces it exactly once. (Practice Q2, Q4)
- I can find the area of a loop of a curve given as r² = g(θ). (Practice Q6(b)–(c))
- I can use a calculator to find a polar area when the limits are not exact angles, writing the integral first. (Guide: Worked example 2; Practice Q7)
Reasoning
- I can explain why an integral over 0 to 2π may double-count a region or give a meaningless answer. (Guide: “Choosing the limits”; Practice Q2(A), Q6(d), Q7(c))
- I can handle an inner loop where r < 0, and separate the area of the inner loop from the area of the outer loop. (Practice Q7)
- I can check a polar area against simple bounds, such as discs or sectors with the smallest and largest r. (Guide: Worked examples 1 and 2)
All ticked? Move on to Topic 9.9, Finding the Area of the Region Bounded by Two Polar Curves, or return to the Calculus BC roadmap.