MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Two polar curves are given by r=3+2sinθ and r=4 for 0≤θ≤π. The region of interest is the area that lies between the two curves within this half–plane.
(a) Determine all values of θ in the interval [0,π] at which the two curves intersect.
(b) For each sub-interval determined in part (a), indicate which curve is farther from the origin. Provide a brief justification that relies on algebraic comparison or a sign diagram.
(c) Explain why the total area of the region must be expressed as the sum of two separate definite integrals. Clearly state the integrand (without evaluating the integrals) that would correspond to each sub-interval.