= Solution
Apply the first <Richardson extrapolation> step to the two computed approximations:
$$
\boxed{R_1(1)=\frac{4(9/32)-1/2}{3}=\frac5{24}.}
$$
The exact integral is $I=\int_0^1x^4\,dx=1/5$, so the actual absolute error is
$$
\boxed{\left|\frac5{24}-\frac15\right|=\frac1{120}\approx0.00833<10^{-2}.}
$$
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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