Consider the pair of polar curves and

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the pair of polar curves r=6sin⁡(3θ)r=6\sin(3\theta) and r=3sin⁡(3θ)r=3\sin(3\theta). Region PP consists of all points that lie between the two curves (that is, satisfy 3sin⁡(3θ)≤r≤6sin⁡(3θ)3\sin(3\theta)\le r\le 6\sin(3\theta)) and for which sin⁡(3θ)≥0\sin(3\theta)\ge 0. (a) Determine the smallest positive angle α\alpha such that 6sin⁡(3α)=3sin⁡(3α)6\sin(3\alpha)=3\sin(3\alpha) and sin⁡(3θ)≥0\sin(3\theta)\ge 0 for 0≤θ≤α0\le\theta\le\alpha. Explain how α\alpha is related to a single "sector" of PP. (b) For 0≤θ≤α0\le\theta\le\alpha, write a definite integral that represents the area enclosed between the two curves. (c) Evaluate the integral from part (b) to find the exact area of that single sector. (d) Using geometric reasoning, obtain the total area of PP from your answer in part (c). State clearly the numerical factor you apply and defend its appropriateness. (e) Suppose a different student integrates from 00 to 2π2\pi without partitioning the region. Provide a concise explanation of why your method in parts (a)–(d) offers a more efficient computation.

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