Consider the region that is simultaneously inside the circle and inside the circle
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Consider the region T that is simultaneously inside the circle r=4cosθ and inside the circle r=2, including any parts traced by negative values of r.
(Note: The graph of r=4cosθ passes through the pole and extends into the left half-plane when cosθ is negative.)
(a) Determine all values of θ with −π≤θ≤π at which the two circles intersect.
(b) Explain why the area of T must be partitioned into at least two sub-regions before integration can begin.
(c) Set up, but do not yet evaluate, the complete collection of definite integrals whose sum equals the area of T. Indicate the limits of integration clearly and account for any portions traced by negative values of r. No simplification is required.
(d) Evaluate your integrals from part (c) to find the exact area of T. Then provide a decimal approximation of the area to three significant figures.