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.