Let be the curve and let be the curve

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Let

C_1

be the curve

r=3\sin\theta

and let

C_2

be the curve

r=3\cos\theta

, both defined for

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

. A student considers the region

\mathcal S

lying between the two curves where they overlap. Answer the following questions without setting up or evaluating any integrals. (a) For

\theta=\frac{\pi}{6}

, decide which curve is farther from the origin. Explain your reasoning solely by comparing the two radial values. (b) Determine all values of

\theta

\bigl(0,\tfrac{\pi}{2}\bigr)

for which

C_1

lies outside

C_2

. Provide a clear justification. (c) Suppose an error in measurement causes every

r

–value on

C_1

to be recorded

0.2

units larger than its actual value while the data for

C_2

remain unaffected. Describe qualitatively how this error changes the area of

\mathcal S

, and state whether the new area is larger than, smaller than, or equal to the true area. Support your answer with appropriate reasoning.