MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
An Archimedean spiral is defined by r=θ for 0≤θ≤1.5.
(1) Set up, but do not evaluate, a definite integral whose value equals the area of the region traced out by the spiral on the given interval.
(2) Using Simpson’s Rule with n=6 equal subintervals, approximate the area to three decimal places.
(3) The function r=θ is increasing on [0,1.5]. Based only on this monotonicity, state whether the Simpson’s Rule approximation in part (b) is expected to be an underestimate, an overestimate, or exact. Provide a brief justification.