MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions
Problem
Consider the polar curve defined by r=2+2cosθ, where θ is measured in radians.
A student wishes to compute the exact area enclosed by this curve and wonders how to select the correct bounds of integration.
(a) Identify all values of θ in the interval 0≤θ<2π for which r=0. Explain how these values partition the tracing of the curve.
(b) Describe, without performing any integration, a single continuous interval of θ over which the curve is traced exactly once without overlap.
(c) Determine whether r assumes negative values on your interval from part (b). Explain the geometric effect of negative r on the location of points generated by the curve.
(d) Justify why the interval you selected in part (b) is appropriate for an area calculation that counts each region exactly once.