MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The limaçon r=1+cosθ is traced once as θ varies from 0 to 2π.
(a) Determine the values of θ in [0,2π] at which the curve passes through the pole (origin). Briefly justify your answer.
(b) Using the information from part (a), set up—but do not evaluate—all definite integral(s) necessary to compute the total length of one full trace of the curve. Be sure to account for any sub-intervals on which separate integrals are required.
(c) Evaluate the integral(s) obtained in part (b) to obtain an exact expression for the total length, and then provide a decimal approximation correct to three significant figures.