Consider the curve given by for

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the curve GG given by r=cos⁡θr=\cos\theta for −π2≤θ≤π2-\dfrac{\pi}{2}\le \theta \le \dfrac{\pi}{2}. Let A1A_1 be the area generated as θ\theta varies from −π2-\dfrac{\pi}{2} to 00, and let A2A_2 be the area generated as θ\theta varies from 00 to π2\dfrac{\pi}{2}. Which statement best justifies the relationship between A1A_1 and A2A_2? (a) A1<A2A_1< A_2 because cos⁡θ\cos\theta decreases on the interval −π2≤θ≤π2-\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}. (b) A1>A2A_1> A_2 because the part of the curve traced for negative θ\theta produces an additional region below the xx-axis. (c) A1=A2A_1=A_2 because (cos⁡θ)2(\cos\theta)^2 is an even function of θ\theta. (d) No comparison can be made since rr is negative for some values of θ\theta in the interval.

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