Consider the cardioid and the circle

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

Problem

Consider the cardioid r1(θ)=2+2sin⁡θr_1(\theta) = 2 + 2\sin \theta and the circle r2(θ)=2sin⁡θr_2(\theta) = 2\sin \theta. For 0≤θ≤π0 \le \theta \le \pi the cardioid encloses the circle, forming an annular-type region. (a) Explain why r1(θ)≥r2(θ)r_1(\theta) \ge r_2(\theta) for every θ\theta in the interval 0≤θ≤π0 \le \theta \le \pi. Your explanation should reference the defining equations of the two curves. (b) Write a definite integral that gives the exact area of the region inside the cardioid and outside the circle on 0≤θ≤π0 \le \theta \le \pi. (c) Rewrite the integrand you obtained in part (b) so that it involves only sin⁡θ\sin \theta (no powers higher than the first) by applying an appropriate trigonometric identity. (d) Evaluate the integral to obtain a simplified exact expression for the area of the region.

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