Discussion
Let f(x) be a function which takes the values \(y_{0}, y_{1}, \ldots, y_{n}\) for \(x = x_{0}, x_{1}, \ldots, x_{n}\) respectively and [a, b] be an interval divided into n subintervals of width h such that :
\[ \begin{aligned} &x_{0} = a, x_{1} = x_{0} + n, \ldots, \\ &x_{n} = x_{0} + nh = b. \end{aligned} \]
Then the error in the trapezoidal rule for integration is of order :
Answer: B
The error of the trapezoidal rule is of order \(h^{2}\).
On one subinterval the trapezoidal rule has error \(-\frac{h^{3}}{12} f''(\xi)\). Over the n subintervals of [a, b], with \(nh = b - a\), the errors add up to
\[ E = -\frac{(b - a)h^{2}}{12} f''(\xi) \]
so the composite error is \(O(h^{2})\). The \(h^{3}\) in option C is the error on a single strip, and h alone would describe a first-order rule such as the rectangle rule.
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