Let be a continuous function defined on with for all in the interval
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Let f be a continuous function defined on [1,7] with f′(x)>0 for all x in the interval. A uniform partition of [1,7] into n subintervals of equal length Δx is used.
A student decides to approximate ∫17f(x)dx by using the left–endpoint Riemann sum Ln as well as the right–endpoint sum Rn.
(a) Describe, without performing any numerical calculation, whether Ln is an underestimate or an overestimate of the integral. Provide a clear verbal justification based on the geometric relationship between the rectangles and the graph of f.
(b) Likewise, indicate whether Rn is an underestimate or an overestimate, and justify your conclusion.
(c) For fixed n, compare the absolute errors ∣Ln−∫17f(x)dx∣ and ∣Rn−∫17f(x)dx∣. Which of the two sums is guaranteed to be closer to the exact value, or are they equal in magnitude? Support your answer with reasoning.