Consider the polar curves given by and ,

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

Problem

Consider the polar curves given by r1(θ)=4cos⁡θ r_1(\theta)=4\cos \theta and r2(θ)=1+cos⁡θ r_2(\theta)=1+\cos \theta , 0≤θ≤π2 0\le \theta\le \tfrac{\pi}{2} . A region in the first quadrant is bounded by these two curves and the initial ray θ=0\theta=0. (a) For each value of θ\theta in the interval 0≤θ≤π20\le \theta\le \tfrac{\pi}{2}, indicate which curve, r1r_1 or r2r_2, lies farther from the origin. Support your answer with appropriate inequalities or sign analysis. (b) Describe in words how the relative positions you found in part (a) determine the integrand that must be used when setting up a single definite integral for the area of the shaded region. (c) Suppose the interval is changed to π2≤θ≤π\tfrac{\pi}{2}\le \theta\le \pi. Without performing any calculations, explain qualitatively how the choice of outer and inner curve would change, and predict whether the resulting integral for the area on that interval would use the same integrand or a different one. Justify your prediction.

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