MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Consider the polar curves r=2cosθ and r=2sinθ. The region common to the two curves is a leaf-shaped area that lies in the first quadrant and is bounded by both graphs.
(a) Identify all lines or axes with respect to which this bounded region is symmetric. Support each claim with a brief verbal justification based on the equations of the curves.
(b) Explain how any symmetries you identified in part (a) affect the choice of limits of integration when setting up a definite integral for the area of the region.
(c) State a single interval of length 4π or less that is sufficient for computing the entire area, and indicate the numerical factor by which the resulting integral should be multiplied to obtain the total area.