Exercise · ZW-TRAPEZOIDAL-RULE-008 · ★★☆ Intermediate
If the error of the composite trapezoidal rule with \(n=2\) on \(\int_0^1 x^2\,dx\) is \(0.0417\), what is the approximate error with \(n=4\)?
Choose one option.
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- D. \(0.0104\)
Explanation
Doubling \(n\) halves the step, and since the error is of order \(h^2\), it is divided by 4: \(0.0417/4\approx0.0104\). Half the error would correspond to dividing by 2, not by 4.
Review the concept: Trapezoidal rule
The trapezoidal rule is a quadrature method that approximates the definite integral of a function \(f\) on \([a,b]\) by replacing the curve with the straight line joining \((a,f(a))\) and \((b,f(b))\).
\[\int_a^b f(x)\,dx \approx \frac{h}{2}\left[f(a)+2\sum_{i=1}^{n-1}f(x_i)+f(b)\right]\]
More practice problems: trapezoidal rule
- ZW-TRAPEZOIDAL-RULE-001 What is the step h when applying the composite trapezoidal rule to ∫_0^2 f(x) dx with n=4 subinterv…
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- ZW-TRAPEZOIDAL-RULE-003 What is the coefficient of f(x_i) for an interior node x_i in the composite trapezoidal rule?
- ZW-TRAPEZOIDAL-RULE-004 For which type of functions is the trapezoidal rule exact?
- ZW-TRAPEZOIDAL-RULE-005 What is the order of the error of the composite trapezoidal rule with respect to the step h?
- ZW-TRAPEZOIDAL-RULE-006 Apply the composite trapezoidal rule to ∫_0^2 x^2 dx with n=2. What value do you get?
- ZW-TRAPEZOIDAL-RULE-007 Using the trapezoidal rule with n=4 on ∫_0^1 e^x dx, what is the approximation given nodes f(0)=1,…
- ZW-TRAPEZOIDAL-RULE-009 What value does the trapezoidal rule give with n=1 for ∫_1^3 (1)/(x) dx?