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=3+3sinθ and r=6sinθ for 0≤θ≤π. Let S denote the region that lies inside both curves simultaneously (the overlap of the two regions).
(a) Determine all values of θ in the interval [0,π] for which the curves intersect. Show your work clearly.
(b) For each subinterval of [0,π] determined by the intersection points, indicate which curve has the smaller radius (is closer to the pole). Justify your answer numerically or algebraically.
(c) Set up, without yet evaluating, one or more definite integrals whose total value equals the exact area of the overlap region S.
(d) Evaluate the integral expression(s) from part (c) to obtain the exact area of S. Write your answer in simplest form.