Exercise · ZW-GAUSSIAN-QUADRATURE-010 · ★★★ Advanced
Using the 2-point Gaussian rule after mapping \([0,2]\) to \([-1,1]\), what is \(\int_0^2 x^3\,dx\)? Use \(x=t+1\) with \(t\in[-1,1]\).
Choose one option.
Show answer
- D. \(4\)
Explanation
The scale factor is 1 and f at the shifted nodes, \((1\pm 1/\sqrt{3})^3\), sums to \(4\), matching the exact value because the degree is 3. Skipping the change of variable leads to values such as \(2\) or \(8\).
Review the concept: Gaussian quadrature
Numerical integration method that chooses optimal nodes \(x_i\) and weights \(w_i\) to approximate an integral with the highest possible accuracy for a fixed number of function evaluations.
\[\int_{a}^{b} f(x)\,dx \approx \sum_{i=1}^{n} w_i\, f(x_i)\]
More practice problems: gaussian quadrature
- ZW-GAUSSIAN-QUADRATURE-001 What is the name of the integral approximation that chooses nodes x_i and weights w_i to gain accur…
- ZW-GAUSSIAN-QUADRATURE-002 What are the nodes x_i of an n-point Gaussian quadrature?
- ZW-GAUSSIAN-QUADRATURE-003 In the formula ∑ w_i f(x_i), what do the w_i represent?
- ZW-GAUSSIAN-QUADRATURE-004 On which standard interval are the Gauss-Legendre nodes defined?
- ZW-GAUSSIAN-QUADRATURE-005 Which family of orthogonal polynomials generates the Gauss-Legendre nodes?
- ZW-GAUSSIAN-QUADRATURE-006 Apply the 2-point rule (nodes pm 1/√3, weights 1) to ∫_-1^1 x^2 dx. What do you get?
- ZW-GAUSSIAN-QUADRATURE-007 To map [a,b] onto [-1,1], which change of variable is used?
- ZW-GAUSSIAN-QUADRATURE-008 If the n-point rule is exact up to degree 2n-1, up to what degree is it exact with n=3?