 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/p/p2/c/solution

R_{1} (1) = \frac{4 ( 9/32 ) - 1/2}{3} = \frac{5}{24} .

The exact integral is

I = \int_{0}^{1} x^{4} d x = 1/5

, so the actual absolute error is

\frac{5}{24} - \frac{1}{5} = \frac{1}{120} \approx 0.00833 < 1 0^{- 2} .

Thus the requested strict error bound is verified directly. This first extrapolation of trapezoidal approximations is also the one-panel Simpson's rule value on the three nodes

0, 1/2, 1

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