Let be the rose curve and let be the circle

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

Problem

Let

C_1

be the rose curve

r = 2 \sin 2\theta

and let

C_2

be the circle

r = \sqrt{2}

. The two curves intersect four times in

0 \le \theta \le 2\pi

. The bounded region

T

is inside the circle and simultaneously inside one petal of the rose. (a) Derive all intersection angles in

[0,2\pi)

where

C_1

and

C_2

meet. (b) Identify the specific sub-interval of

\theta

that traces exactly one petal of the rose lying inside the circle. Justify your interval choice. (c) Construct an integral (or sum of integrals, if necessary) that gives the exact area of

T

. (d) Evaluate your expression from part (c) to obtain the exact area. (e) Suppose the circle is replaced by

r = k

with

k>0

. Determine, in terms of

k

, the range of

k

values for which the bounded region described in parts (a)–(d) still exists, and justify your answer.

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