MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
Consider the rose curve defined by r=asin(2θ) for 0≤θ≤2π, where a>0 is a constant.
(a) Describe geometrically why the curve, although it meets the pole at both endpoints of the interval, encloses a finite length on 0≤θ≤2π. Your answer should reference the behavior of r near the pole.
(b) Without evaluating a definite integral, explain how symmetry of the curve on the given interval could be exploited to simplify the computation of its length.
(c) Identify any substitution of the form u=g(θ) that would convert the integrand in the arc-length integral to an algebraic or rational expression in u. Justify why your proposed substitution is effective given the structure of sin(2θ). (No computation of the resulting integral is required.)