MATHEMATICS · Subjective · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
For the polar curve given by r(θ)=1+2sinθ on 0≤θ≤π, consider the portion of the curve traced once counter-clockwise beginning at θ=0.
(a) Write a definite integral whose value equals the length of this portion of the curve.
(b) Using n=6 subintervals of equal width, approximate the length from part (a) with a right-endpoint Riemann sum. Express your answer correct to three decimal places.
(c) A calculus student claims that doubling the number of subintervals will reduce the absolute error of the approximation in part (b) by a factor of approximately four. Determine whether the claim is always true for this curve when the right-endpoint Riemann sum is used. Provide mathematical justification for your conclusion.