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=3sinθ and r=2cosθ. Consider the region that is common to the interiors of both curves (that is, the region that lies simultaneously inside both curves).
Answer the following parts.
(1) Solve 3sinθ=2cosθ for θ in the interval 0≤θ≤2π, and list the solutions in increasing order.
(2) On the interval found in part (a), determine which curve forms the outer boundary of the common region. Provide reasoning that references the values of r for each curve.
(3) Construct an integral (or a sum of integrals) that gives the exact area of the region common to the interiors of both curves over 0≤θ≤2π. Do not evaluate yet.
(4) Evaluate the expression from part (c) to obtain the exact area of the entire common region (include any necessary symmetry considerations).