A cardioid-shaped curve and a circle are given by and
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A cardioid-shaped curve and a circle are given by r=1+cosθ and r=3, respectively. Region S is the set of points that satisfy 1+cosθ≤r≤3.
(a) Find all values of θ between 0 and 2π at which the two curves intersect.
(b) Decide whether the region S possesses any symmetry that can be used to shorten the interval of integration for its area. State your conclusion and justify it with reference to the defining equations.
(c) Construct a definite integral (or sum of integrals) that equals the exact area of S. Clearly indicate all limits of integration.
(d) Evaluate your integral(s) to obtain the exact area of S. Simplify your answer as much as possible.