The concentration of a dissolved mineral (in milligrams per liter) at various points…
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
The concentration of a dissolved mineral (in milligrams per liter) at various points along a 400-meter stretch of a river is modeled by a continuous function C(x), where x is the distance in meters downstream from the sampling station located at x=0. Measurements of C were taken at five equally spaced locations, yielding the data in the table below.
Distance x (m): 0 100 200 300 400
Concentration C(x) (mg/L): 16.4 18.7 20.1 21.0 21.6
Throughout the entire interval 0≤x≤400, the function C is increasing and concave down.
(a) Provide a practical interpretation, in the context of the river, of the quantity 4001∫0400C(x)dx.
(b) Without performing a detailed computation, state whether the Trapezoidal Rule, applied to the data in the table, will produce an overestimate or an underestimate of the average concentration found in part (a). Give a clear justification based on properties of C. No numerical calculation is required.
(c) Suppose instead that the Midpoint Rule with the same subinterval endpoints were used. Explain how the sign of the approximation error would change and provide reasoning for your conclusion.