Let be the curve and let be the curve

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

Problem

Let

C_1

be the curve

r=2+2\sin\theta

and let

C_2

be the curve

r=2\sin\theta

. Region

R

is the set of points that lie inside

C_1

and outside

C_2

. (a) Solve

2+2\sin\theta = 2\sin\theta

for

\theta

0\le\theta\le 2\pi

to find all intersection angles of

C_1

and

C_2

. (b) Using the equations of

C_1

and

C_2

, establish whether

R

is symmetric with respect to the line

\theta=\dfrac{\pi}{2}

. Provide a clear argument. (c) Write a definite integral that computes the area of the portion of

R

lying on one side of the line you discussed in part (b), and indicate the factor needed to obtain the entire area of

R

. (d) Evaluate the integral from part (c) to find the exact total area of

R

. (e) Suppose instead that the area were computed by integrating directly from

0

2\pi

without any reduction of limits. Describe one advantage, both algebraic and numerical, of the approach you used in parts (c)–(d).