For a differentiable function on

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

Problem

For a differentiable function hh on [a,b][a,b], two Riemann sums are formed: one using left endpoints LNL_N and the other using right endpoints RNR_N, each based on NN equal subintervals. (a) Explain conceptually why lim⁡N→∞LN\displaystyle \lim_{N\to\infty}L_N and lim⁡N→∞RN\displaystyle \lim_{N\to\infty}R_N must be equal when hh is continuous on [a,b][a,b]. (b) Let h(x)=sin⁡xh(x)=\sin x on [0,π2]\left[0,\tfrac{\pi}{2}\right]. Write both LNL_N and RNR_N explicitly in sigma notation, then prove that their common limit equals ∫0π/2sin⁡x dx\displaystyle \int_{0}^{\pi/2}\sin x\,dx. Your proof must include a justification that the difference ∣RN−LN∣|R_N-L_N| tends to zero as N→∞N\to\infty.

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