Consider on the interval

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

Problem

Consider f(x)=ln⁡(1+x)f(x)=\ln(1+x) on the interval [0,2][0,2]. Divide [0,2][0,2] into nn equal subintervals and derive an explicit formula for the approximation RnR_n of ∫02f(x) dx\displaystyle\int_0^2f(x)\,dx that uses the right endpoints of the subintervals. Evaluate lim⁡n→∞Rn\displaystyle\lim_{n\to\infty}R_n and demonstrate that this limit equals the exact value of the definite integral. Determine the least integer nn such that ∣Rn−∫02f(x) dx∣<0.01\left|R_n-\int_0^2f(x)\,dx\right|<0.01. Justify your answer. (1) (a) (2) (b) (3) (c)

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