A curve given in Cartesian form by the equation and the polar circle bound a finite…
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A curve given in Cartesian form by the equation x2+y2=4x and the polar circle r=3 bound a finite region U lying between them.
(a) Rewrite the Cartesian equation in polar coordinates and simplify.
(b) Solve for all angles θ where the rewritten curve intersects the circle r=3.
(c) Determine which curve serves as the outer boundary on each sub-interval you found in part (b). Provide justification by comparing radial distances.
(d) Develop an exact integral expression for the area of U and evaluate it.
(e) Discuss how the area would change if the circle were replaced by r=c, where c>0. Identify any critical value of c that alters the qualitative nature of the bounded region and justify your conclusion.