Let and , traced for one full revolution . A finite region is enclosed by alternating…
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Let r1=1+2sinθ and r2=2cosθ, traced for one full revolution 0≤θ≤2π. A finite region is enclosed by alternating portions of these two curves.
(a) Derive the equation that must be solved to locate all intersection angles of the curves on [0,2π].
(b) Using either algebraic methods or numerical approximation, list all intersection angles in ascending order accurate to three decimal places.
(c) Provide a reasoned argument—supported by a sign chart, inequalities, or a clearly labeled sketch—that identifies which curve is farther from the origin on each open interval determined in part (b).
(d) Construct the full piece-wise integral expression for the total enclosed area and evaluate it to three decimal places.
(e) One of your definite integrals may return a negative value before simplification. Explain why the final area reported in part (d) remains positive despite this sign issue.