A region is inside the cardioid and outside the circle

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

Problem

A region

R

is inside the cardioid

r=2+2\cos\theta

and outside the circle

r=2

. The curves intersect in the first and fourth quadrants. Answer the following questions about

R

. (a) Solve

2+2\cos\theta = 2

to find all values of

\theta

[-\pi,\pi]

where the two curves intersect. (b) Describe, without calculation, why the area of

R

cannot be represented by a single definite integral of the form

\displaystyle\tfrac12\int_{\alpha}^{\beta} r^2\,d\theta

. In your explanation, refer to the behavior of

r

on the interval and the geometric shape of

R

. (No numerical answer required.) (c) Write (but do not evaluate) a complete piecewise expression consisting of one or more definite integrals whose sum equals the exact area of

R

. Clearly indicate the limits of integration and the integrand for each sub-region you define.

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