Question 4.7.1: Using the Midpoint Rule Write out the Midpoint Rule approxim...
Using the Midpoint Rule
Write out the Midpoint Rule approximation of \int_0^1 3 x^2 d x with n = 4.
For n = 4, the regular partition of the interval [0, 1] is x0 = 0, x_1=\frac{1}{4}, x_2=\frac{1}{2}, x_3=\frac{3}{4} and x4 = 1, The midpoints are then c_1=\frac{1}{8}, c_2=\frac{3}{8}, c_3=\frac{5}{8} and c_4=\frac{7}{8}.
\left[f\left(\frac{1}{8}\right)+f\left(\frac{3}{8}\right)+f\left(\frac{5}{8}\right)+f\left(\frac{7}{8}\right)\right]\left(\frac{1}{4}\right)=\left(\frac{3}{64}+\frac{27}{64}+\frac{75}{64}+\frac{147}{64}\right)\left(\frac{1}{4}\right)= \frac{252}{256}=0.984375.
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