A particle moves along a line with position function (in meters) for

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A particle moves along a line with position function s(t)=ln⁡ts(t)=\ln t (in meters) for 1≤t≤41\le t\le4, where tt is measured in seconds. Equal subintervals of length Δt=0.5\Delta t=0.5 are used to approximate ∫14ln⁡t dt\displaystyle \int_{1}^{4}\ln t\,dt, the net displacement of the particle. (a) Compute the Left–Riemann sum L6L_6 and the Right–Riemann sum R6R_6 that approximate the integral using the six subintervals determined by t0=1,  t1=1.5,  t2=2,  …,  t6=4t_0=1,\;t_1=1.5,\;t_2=2,\;\dots,\;t_6=4. Present each sum as a numerical value accurate to three decimal places. (b) Evaluate ∫14ln⁡t dt\displaystyle \int_{1}^{4}\ln t\,dt exactly, and give a decimal approximation to three decimal places. (c) Based on the behavior of ln⁡t\ln t on the interval, determine which of the two Riemann sums in part (a) is closer to the exact integral. Quantify the absolute error of each approximation and comment on whether each sum over–estimates or under–estimates the displacement.

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