Discussion
The order of magnitude of the error in estimating the integral \(\int_{0}^{1}(x^{3} + 2x^{2} + 4x + 9)\,dx\) using Simpson's rule with step size 0.1 is :
Answer: A
The error is 0.
The error term of Simpson's rule is proportional to the fourth derivative of the integrand:
\[ E = -\frac{(b-a)h^{4}}{180} f^{(4)}(\xi) \]
The integrand \(x^{3} + 2x^{2} + 4x + 9\) is a cubic, so \(f^{(4)} = 0\) everywhere and Simpson's rule gives the exact value whatever the step size. Although the rule is built by fitting parabolas, it integrates cubics exactly as well, because the cubic error cancels by symmetry on each pair of strips.
The options \(10^{-3}\) to \(10^{-5}\) are the sizes one would expect from \(h^{4} = 10^{-4}\) for a general smooth function; they do not apply to a polynomial of degree three or less.
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