MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Let f be a positive, continuous function defined on [a,b]. Assume throughout that the interval is partitioned into n equal subintervals of width Δx=nb−a. In responding to each part, provide clear verbal reasoning rooted in the behavior of f on the stated interval; no numerical computation is required.
(a) Suppose f is strictly increasing on [a,b]. Determine whether the Left–Riemann sum Ln provides an over–estimate or an under–estimate of ∫abf(x)dx. Explain your conclusion.
(b) Suppose instead that f is strictly decreasing on [a,b]. Decide whether Rn, the Right–Riemann sum, is an over–estimate or an under–estimate of the integral. Justify your answer.
(c) Now assume f is strictly increasing and concave up on [a,b]. Compare the Midpoint sum Mn and the Trapezoidal sum Tn to the exact value of the integral. State which of the two sums is larger and which (if any) is guaranteed to lie on the same side of the exact integral as Ln. Support your reasoning with a discussion of concavity.
(d) Finally, consider f strictly decreasing and concave down on [a,b]. Rank Ln,Mn,Rn, and Tn from least to greatest. Provide a coherent justification that refers explicitly to both monotonicity and concavity.