Mathematics

Problem

Consider the function

h(x)=\sin\bigl(x^{2}\bigr)

on the interval

[0,1]

. (a) Partition the interval into four equal subintervals and list the midpoint of each subinterval. (b) Form the midpoint Riemann sum

M_{4}

that approximates

\displaystyle \int_{0}^{1} h(x)\,dx

, and evaluate it to four decimal places. Calculations may be performed with a graphing calculator. (c) The exact value of

\displaystyle \int_{0}^{1} h(x)\,dx

is approximately

0.3103

. Compute the percent error of the approximation obtained in part (b) and comment on the effectiveness of the midpoint rule in this situation. (a) — see stem

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