MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The polar graph of r=6cos2θ forms a four-petaled rose, while the circle r=3 encloses a disk centered at the pole. Let T be the set of all points that are inside the circle and outside the rose.
(a) Show that the curves intersect exactly eight times in 0≤θ<2π and list these intersection angles in increasing order.
(b) One of the intersection angles found in part (a) is θ=6π. Determine the two consecutive intersection angles between which the region T occupies the greatest angular width inside the first quadrant.
(c) Set up (but do not evaluate) an expression involving one or more integrals whose value equals the area of T.
(d) Evaluate the expression from part (c) to obtain the exact area of T.