Discussion
P (0, 3), Q (0.5, 4), and R (1, 5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson’s rule within limits \(x = 0\) and \(x = 1\) for the curve. The difference between the two results will be
Answer: A
The difference is 0.
The three points (0, 3), (0.5, 4) and (1, 5) lie on the straight line y = 3 + 2x. Both rules are exact for straight lines:
\[ \begin{aligned} \text{Trapezoidal} &= \tfrac{0.5}{2}[3 + 2(4) + 5] = 4 \\ \text{Simpson} &= \tfrac{0.5}{3}[3 + 4(4) + 5] = 4 \end{aligned} \]
Both give 4, the exact area, so they agree. Simpson's rule differs from the trapezoidal rule only when the data have curvature; 0.25, 0.5 and 1 would require a curve through the points.
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