MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Let C1 be the curve r=sinθ and let C2 be the curve r=sin(2θ), both plotted for 0≤θ≤π. Denote by S the region that lies inside both C1 and C2.
(a) Find all values of θ in [0,π] at which the two curves intersect.
(b) Using your results from part (a), identify on each subinterval of [0,π] which of the two curves is farther from the pole (i.e., which curve is the outer boundary of S).
(c) Construct a piecewise definite integral (or sum of integrals) whose value equals the area of S. Clearly indicate the bounds associated with each piece.
(d) Evaluate the integral expression from part (c) to obtain the exact area of S.