MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The circle C is described by r=2. The spiral S is described by r=θ. For the portion of the spiral that first intersects the circle beyond θ=0, a region is trapped between the two curves.
(a) Find the positive value of θ (other than θ=0) at which the spiral and the circle intersect for the first time.
(b) Formulate a single definite integral whose value equals the area inside the circle r=2 but outside the spiral r=θ, bounded by the rays θ=0 and the intersection value found in part (a).
(c) Evaluate the integral from part (b) to three decimal places.