MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A limacon is given by the polar equation r=1+2cosθ. Because r takes on negative values for certain θ, the curve possesses an inner loop.
(a) Solve 1+2cosθ=0 on 0≤θ<2π to identify the angular positions separating the inner loop from the outer part of the curve.
(b) Set up, but do not evaluate, a definite integral that represents the exact area of the inner loop. Include correct bounds and integrand.
(c) Evaluate the integral from part (b) to obtain the exact area of the inner loop.
(d) Determine the exact area enclosed by the entire limacon and then compute the exact area that lies outside the inner loop but inside the outer boundary.