Let and . A shaded region is bounded on the outside by and on the inside by in the…

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

Problem

Let

r_{1}(\theta)=2\bigl(1+\cos\theta\bigr)

and

r_{2}(\theta)=4\cos\theta

. A shaded region

R

is bounded on the outside by

r_{1}

and on the inside by

r_{2}

in the first quadrant. (a) Determine all values of

\theta

between

0

and

\tfrac{\pi}{2}

for which the two curves intersect. Show analytical work leading to your result. (b) Using the bounds found in part (a), set up—but do not yet evaluate—a single definite integral whose value equals the exact area of region

R

. (c) Evaluate the integral obtained in part (b) and express the area of

R

in exact form.

Related theory

Polar Graphs
Trigonometric Functions
Polar Functions

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