Find the integral I = integral of and limit is -1 to 1 e^-x^2 cos xdx using two-point Gauss Legendre rule. Marks 

To evaluate the integral \( I = \int_{-1}^{1} e^{-x^2} \cos(x) \, dx \) using the two-point Gauss-Legendre quadrature rule, we'll follow these steps:

### 1. **Gauss-Legendre Quadrature Rule:**
For a two-point Gauss-Legendre quadrature, the integral over the interval \([-1, 1]\) is approximated by:

\[
\int_{-1}^{1} f(x) \, dx \approx \sum_{i=1}^{2} w_i \cdot f(x_i)
\]

where:
- \( x_1 = -\frac{1}{\sqrt{3}} \) and \( x_2 = \frac{1}{\sqrt{3}} \) are the roots of the Legendre polynomial \( P_2(x) \).
- \( w_1 = w_2 = 1 \) are the weights.

### 2. **Evaluate the Function at the Quadrature Points:**

We need to evaluate the function \( f(x) = e^{-x^2} \cos(x) \) at the points \( x_1 \) and \( x_2 \):

- For \( x_1 = -\frac{1}{\sqrt{3}} \):

\[
f\left(-\frac{1}{\sqrt{3}}\right) = e^{-\left(-\frac{1}{\sqrt{3}}\right)^2} \cos\left(-\frac{1}{\sqrt{3}}\right) = e^{-\frac{1}{3}} \cos\left(-\frac{1}{\sqrt{3}}\right)
\]

Since \( \cos(x) \) is an even function, \( \cos(-x) = \cos(x) \):

\[
f\left(-\frac{1}{\sqrt{3}}\right) = e^{-\frac{1}{3}} \cos\left(\frac{1}{\sqrt{3}}\right)
\]

- For \( x_2 = \frac{1}{\sqrt{3}} \):

\[
f\left(\frac{1}{\sqrt{3}}\right) = e^{-\left(\frac{1}{\sqrt{3}}\right)^2} \cos\left(\frac{1}{\sqrt{3}}\right) = e^{-\frac{1}{3}} \cos\left(\frac{1}{\sqrt{3}}\right)
\]

### 3. **Apply the Gauss-Legendre Rule:**

Now, using the two-point Gauss-Legendre rule:

\[
I \approx w_1 \cdot f(x_1) + w_2 \cdot f(x_2)
\]

Substituting the weights and function values:

\[
I \approx 1 \cdot e^{-\frac{1}{3}} \cos\left(\frac{1}{\sqrt{3}}\right) + 1 \cdot e^{-\frac{1}{3}} \cos\left(\frac{1}{\sqrt{3}}\right)
\]

This simplifies to:

\[
I \approx 2 \cdot e^{-\frac{1}{3}} \cos\left(\frac{1}{\sqrt{3}}\right)
\]

### 4. **Final Answer:**
The integral is approximately:

\[
I \approx 2 \cdot e^{-\frac{1}{3}} \cos\left(\frac{1}{\sqrt{3}}\right)
\]

This is the approximate value of the integral \( I \) using the two-point Gauss-Legendre quadrature rule.

Would you like more details or have any further questions?

Here are some related questions for further exploration:

1. How does the result compare if a four-point Gauss-Legendre quadrature rule is used?
2. What are the error bounds for this approximation?
3. How would the integral be computed using Simpson's rule?
4. How does the accuracy of Gauss-Legendre quadrature compare with other numerical integration methods?
5. What is the significance of the Gauss-Legendre quadrature rule in computational mathematics?

**Tip:** For integrals of even functions over symmetric intervals, Gauss-Legendre quadrature can provide highly accurate results with fewer points.

Learn how to evaluate the integral ∫_{-1}^{1} e^{-x^2} cos(x) dx using the two-point Gauss-Legendre quadrature rule. This method approximates the integral using specific points and weights, ensuring accurate results for even functions over symmetric intervals. Suitable for undergraduate-level students and anyone interested in numerical integration methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation