For a polar curve whose equation produces negative -values on part of the interval
MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
For a polar curve whose equation produces negative r-values on part of the interval α≤θ≤β, a student wants to compute the exact area of the region traced once as θ increases from α to β. Which procedure best guarantees that the calculated area is correct?
(a) Integrate 21r2 only over the portion of the interval where r>0 and then double the result, since the part where r<0 is a reflected copy.
(b) Integrate 21r2 over the entire interval α≤θ≤β; squaring r removes the sign, so the integral yields the desired area.
(c) Integrate r over α≤θ≤β and square the value you obtain to make the area positive.
(d) Replace r with ∣r∣ in the integrand and divide the final answer by 2 to correct for the absolute value.