Two polar curves are given by and

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Two polar curves are given by

r_{1}=2\cos\theta

and

r_{2}=4\cos\theta

. Let

Q

be the region in the first quadrant that lies inside

r_{2}

and outside

r_{1}

. (a) Convert the equations of both curves to rectangular form. State the center and radius of each corresponding circle. (b) Determine the exact values of

\theta

that locate the points of intersection of the two curves in the first quadrant. Justify your answer algebraically. (c) Compute the exact area of

Q

by setting up and evaluating a definite integral in polar coordinates. (d) Verify your result from part (c) by expressing the area of

Q

as the difference between two circular sectors in rectangular geometry. Show all supporting work.

Related theory

Polar Coordinates
Trigonometric Functions
Polar Functions
Trigonometric Applications

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