Discussion
Consider the definite integral
\[ \int_{1}^{2} (4x^{2} + 2x + 6)\,dx. \]
Let \(I_{e}\) be the exact value of the integral. If the same integral is estimated using Simpson's rule with 10 equal subintervals, the value is \(I_{s}\). The percentage error is defined as \(e = 100 \times (I_{e} - I_{s})/I_{e}\). The value of \(e\) is
Answer: D
The percentage error is 0.
Simpson's 1/3 rule fits a parabola through each pair of subintervals, and its error term involves the fourth derivative of the integrand. For a polynomial of degree three or less the fourth derivative is zero, so the rule is exact, whatever the number of subintervals.
The integrand \(4x^{2} + 2x + 6\) is a quadratic, so \(I_{s} = I_{e}\) and \(e = 0\). The exact value is
\[ \begin{aligned} I_{e} &= \left[\tfrac{4x^{3}}{3} + x^{2} + 6x\right]_{1}^{2} \\ &= \tfrac{28}{3} + 3 + 6 = \tfrac{55}{3} \end{aligned} \]
and Simpson's rule returns the same 55/3. The non-zero options would describe a rule that is not exact for quadratics, such as the trapezoidal rule.
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