MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A polar curve is defined for 0≤θ≤π by r=2+2sinθ. Without evaluating any definite integrals, answer the following.
(a) Rewrite the curve parametrically as x(θ) and y(θ) for 0≤θ≤π.
(b) Compute dθdx and dθdy in terms of θ.
(c) Construct a definite integral in terms of θ whose value equals the length of the curve for 0≤θ≤π. Include all limits of integration but do not simplify the integrand.
(d) Briefly explain whether the integral you obtained in part (c) can be expressed using elementary functions. Support your conclusion with appropriate reasoning, but no computation is required.