The limaçon and the cardioid bound a finite region
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The limaçon C1:r=5+3cosθ and the cardioid C2:r=4(1−cosθ) bound a finite region U. Because C1 and C2 intersect at angles that cannot be expressed in closed form, numerical methods are required.
(a) Use a graphing calculator to approximate all intersection angles of C1 and C2 in 0≤θ≤2π, accurate to three decimal places.
(b) Explain why any extraneous numerical solutions produced by a calculator solver must be discarded, citing geometric reasoning involving the two curves.
(c) Construct a sum of definite integrals that exactly represents the area of U. Clearly indicate the limits of integration corresponding to each subregion.
(d) Use technology to evaluate the integrals in part (c) and report the area of U to three decimal places.
(e) Interpret the numerical result of part (d) in the context of the graphs of C1 and C2, commenting on the relative sizes of the two curves over the interval of intersection.