The Gauß–Legendre quadrature 𝐺(𝑓 ) (−1,1) with 3 nodes on the interval [−1,1] is given by the nodes: 𝑥_0 = −sqrt(3/5), 𝑥_1 = 0, 𝑥_2 = +sqrt(3/5), and the weights: 𝑤_0 = 5/9, 𝑤_1 = 8/9, 𝑤_2 = 5/9.

Transfer this quadrature rule to an arbitrary interval (𝑎, 𝑏) to obtain an approximation 𝐺(𝑓 ) (𝑎, 𝑏) ≈ ∫ ^𝑏_𝑎 𝑓(𝑥) 𝑑𝑥

To transfer the Gauß–Legendre quadrature rule from the standard interval \([-1, 1]\) to an arbitrary interval \([a, b]\), you must perform a change of variables to map the nodes and weights accordingly. The standard process involves scaling and shifting the nodes and adjusting the weights to account for the length of the new interval.

### Step 1: Change of Variables
The Gauß–Legendre quadrature rule is designed for the interval \([-1, 1]\). For an arbitrary interval \([a, b]\), you apply the following linear transformation to the variable \(x\):

\[
x = \frac{b - a}{2} \cdot t + \frac{b + a}{2}
\]

where \(t\) is the variable in the interval \([-1, 1]\) (which corresponds to the original nodes \(x_i\)), and \(x\) is the transformed variable in the interval \([a, b]\).

### Step 2: Adjust the Nodes
The nodes \(x_i\) in the interval \([a, b]\) are given by:

\[
x_i = \frac{b - a}{2} \cdot t_i + \frac{b + a}{2}
\]

where \(t_i\) are the original nodes from \([-1, 1]\).

For the given problem, the original nodes are:

\[
t_0 = -\sqrt{\frac{3}{5}}, \quad t_1 = 0, \quad t_2 = \sqrt{\frac{3}{5}}
\]

Thus, the new nodes \(x_i\) in the interval \([a, b]\) become:

\[
x_0 = \frac{b - a}{2} \cdot \left(-\sqrt{\frac{3}{5}}\right) + \frac{b + a}{2}
\]
\[
x_1 = \frac{b - a}{2} \cdot 0 + \frac{b + a}{2} = \frac{b + a}{2}
\]
\[
x_2 = \frac{b - a}{2} \cdot \left(\sqrt{\frac{3}{5}}\right) + \frac{b + a}{2}
\]

### Step 3: Adjust the Weights
The weights \(w_i\) also need to be scaled by the factor \(\frac{b - a}{2}\) because the length of the interval has changed. The new weights \(w_i^{\prime}\) for the interval \([a, b]\) are:

\[
w_i^{\prime} = \frac{b - a}{2} \cdot w_i
\]

For the given problem, the original weights are:

\[
w_0 = \frac{5}{9}, \quad w_1 = \frac{8}{9}, \quad w_2 = \frac{5}{9}
\]

Thus, the new weights for the interval \([a, b]\) are:

\[
w_0^{\prime} = \frac{b - a}{2} \cdot \frac{5}{9}
\]
\[
w_1^{\prime} = \frac{b - a}{2} \cdot \frac{8}{9}
\]
\[
w_2^{\prime} = \frac{b - a}{2} \cdot \frac{5}{9}
\]

### Step 4: Form the Quadrature Approximation
The Gauß–Legendre quadrature rule on the interval \([a, b]\) is given by:

\[
\int_a^b f(x) \, dx \approx \sum_{i=0}^{2} w_i^{\prime} f(x_i)
\]

Substitute the new nodes \(x_i\) and the new weights \(w_i^{\prime}\) into the formula to approximate the integral over \([a, b]\).

Would you like to see more details or have any questions?

Here are five follow-up questions:

1. How does the change of variables affect the nodes in more complex quadrature rules?
2. Can the quadrature rule be extended to intervals with infinite bounds, such as \((-\infty, \infty)\)?
3. What is the error term for the Gauß–Legendre quadrature with 3 nodes?
4. How does the choice of the number of nodes affect the accuracy of the quadrature?
5. Can we use the same approach to transfer other quadrature rules, such as Simpson's rule, to different intervals?

**Tip**: When using the Gauß–Legendre quadrature rule, increasing the number of nodes (points) improves the accuracy of the approximation for smooth functions.

The Gauß–Legendre quadrature 𝐺(𝑓 ) (−1,1) with 3 nodes on the interval [−1,1] is given by the nodes: 𝑥_0 = −sqrt(3/5), 𝑥_1 = 0, 𝑥_2 = +sqrt(3/5), and the weights: 𝑤_0 = 5/9, 𝑤_1 = 8/9, 𝑤_2 = 5/9. Transfer this quadrature rule to an arbitrary interval (𝑎, 𝑏) to obtain an approximation 𝐺(𝑓 ) (𝑎, 𝑏) ≈ ∫ ^𝑏_𝑎 𝑓(𝑥) 𝑑𝑥.

Learn how to transfer the Gauß–Legendre quadrature rule from the standard interval [-1,1] to any arbitrary interval (a, b) using a simple change of variables. This solution involves scaling nodes and adjusting weights. Suitable for students and professionals studying numerical integration in undergraduate-level courses.

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