The function is increasing and concave down on the closed interval
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
The function h is increasing and concave down on the closed interval [2,5]. A student considers approximating ∫25h(x)dx with equally spaced sample points.
(a) Explain whether a trapezoidal sum that uses the endpoints of each subinterval yields an overestimate or an underestimate of ∫25h(x)dx. Provide a concise reason that relies on geometric reasoning about the graph of h.
(b) Explain whether a midpoint Riemann sum with the same partition yields an overestimate or an underestimate of ∫25h(x)dx.
(c) Suppose the exact integral equals 7.8. Decide for each of the following pairs of numerical estimates whether the pair is possible or impossible, and justify your conclusions.
(i) T=7.3 (trapezoidal) and M=8.1 (midpoint)
(ii) T=8.0 (trapezoidal) and M=7.5 (midpoint)