MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Consider the two polar curves r=cosθ and r=cos(2θ). Let T be the collection of all regions that are simultaneously inside both curves for 0≤θ<2π.
(a) Determine every value of θ in [0,2π) at which the curves intersect.
(b) List the angular subintervals on which the set of points common to both curves forms a bounded region. Give a brief justification for each interval you list.
(c) Choose the bounded region that lies in the first quadrant and write a complete definite integral (or sum of integrals) that computes its area. Make sure the limits of integration correspond to the intersections found in part (a).
(d) Evaluate the integral you wrote in part (c) to obtain an exact value for the area of the chosen region.