MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Consider the polar curves C1:r=2+cosθ and C2:r=3sinθ. The curves bound a finite region that lies entirely in the first quadrant.
(a) Determine all values of θ in [0,2π] where C1 and C2 intersect.
(b) For each sub-interval you identify in part (a), indicate which curve forms the outer boundary of the region and which curve forms the inner boundary. Justify your conclusion without evaluating any integrals.
(c) Write, but do not evaluate, the integral expression that represents the exact area of the region bounded by C1 and C2 in the first quadrant.
(d) Explain how the sign of the integrand in part (c) would change if the inner and outer curves were mistakenly reversed, and describe the geometric consequence of this error.