The circle and the limaçon enclose a shaded region lying outside the limaçon but…
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The circle r=3 and the limaçon r=1+2sinθ enclose a shaded region S lying outside the limaçon but inside the circle.
(a) Solve for all angles θ that describe points common to both curves.
(b) Using the angles from part (a), write a single integral or a sum of integrals that represents the exact area of S.
(c) Evaluate the integral(s) from part (b) to obtain the exact area of S.
(d) Interpret your result in terms of the relative sizes of the two curves and comment on whether the limaçon ever becomes the outer boundary for some values of θ. Provide justification.