Let be the region in the first quadrant bounded by the polar curve
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Let S be the region in the first quadrant bounded by the polar curve r=3sinθ, the ray θ=6π, and the circle r=2.
(a) Sketch S on the polar plane, labeling the points where the three boundaries meet.
(b) Show algebraically that the circle and the sine curve intersect exactly once in the first quadrant and find the corresponding value of θ.
(c) Express the area of S as a sum of definite integrals. Be sure each integral matches a specific sub-region on your sketch.
(d) Evaluate the integrals in part (c) to obtain the exact area of S.
(e) Give a decimal approximation of the area of S, correct to three decimal places.