Two bounded polar regions are defined as follows

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

Problem

Two bounded polar regions are defined as follows. • Region S is enclosed by the curve

r = 3

and the rays

\theta = 0

and

\theta = \frac{\pi}{2}

. • Region T is enclosed by the curve

r = 4\sin\theta

and the same two rays. Assume all angles are measured in radians and

0 \leq \theta \leq \frac{\pi}{2}

. (a) Determine every value of

\theta

in the interval

0 \leq \theta \leq \frac{\pi}{2}

at which the two curves intersect. (b) Write, but do NOT evaluate, an integral expression for the area of the portion of

S

that lies outside

T

. (c) Evaluate the integral you wrote in part (b) to obtain the exact area (in square units). (d) Calculate the percentage difference between the total areas of regions

S

and

T

. (Clarification: the percentage difference is defined as the absolute difference between the areas divided by the average of the two areas, multiplied by 100.)

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