A circle with polar equation and a cardioid with polar equation enclose a shaded…
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A circle with polar equation r=2 and a cardioid with polar equation r=4cosθ enclose a shaded region that lies entirely in the first quadrant (where 0≤θ≤2π). The region is outside the circle and inside the cardioid.
Answer the following parts, expressing all exact values in terms of π where appropriate.
(1) Determine the exact θ–values that bound the portion of the circle lying inside the cardioid in the first quadrant. Justify your answer algebraically.
(2) Using your result from part (a), write an integral (or sum of integrals, if necessary) that represents the area of the shaded region.
(3) Evaluate the integral(s) from part (b) to obtain the exact area of the shaded region.