Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Please help with n=32
**Using Simpson's Rule for Numerical Integration** In this section, we'll apply Simpson's Rule to approximate the integral of the function \( \int_0^8 (3x^5 - 2x^3) \, dx \), with the exact value being 129,024. We'll provide a table showing approximations and errors for different segment values \( n = 4, 8, 16, \) and \( 32 \). **Table of Approximations and Errors** | \( n \) | \( T(n) \) | \( S(n) \) | Absolute Error in \( T(n) \) | Absolute Error in \( S(n) \) | |---------|-----------|----------|-------------------------|-------------------------| | 4 | 149,120 | — | 20,096 | — | | 8 | 134,096 | 129,088 | 5,072 | 64 | | 16 | 130,295 | 129,028 | 1,271 | 4 | | 32 | 129,341.9 | 129,024.25 | 317.9 | 0.25 | **Explanation:** - \( T(n) \) refers to the approximation using the Trapezoidal Rule. - \( S(n) \) refers to the approximation using Simpson's Rule. - The absolute errors for each method are calculated by comparing the approximation with the exact integral value \( 129,024 \). Simpson's Rule generally provides a more accurate approximation, as evidenced by smaller errors compared to the Trapezoidal Rule, particularly as \( n \) increases.
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Transcribed Image Text:**Using Simpson's Rule for Numerical Integration** In this section, we'll apply Simpson's Rule to approximate the integral of the function \( \int_0^8 (3x^5 - 2x^3) \, dx \), with the exact value being 129,024. We'll provide a table showing approximations and errors for different segment values \( n = 4, 8, 16, \) and \( 32 \). **Table of Approximations and Errors** | \( n \) | \( T(n) \) | \( S(n) \) | Absolute Error in \( T(n) \) | Absolute Error in \( S(n) \) | |---------|-----------|----------|-------------------------|-------------------------| | 4 | 149,120 | — | 20,096 | — | | 8 | 134,096 | 129,088 | 5,072 | 64 | | 16 | 130,295 | 129,028 | 1,271 | 4 | | 32 | 129,341.9 | 129,024.25 | 317.9 | 0.25 | **Explanation:** - \( T(n) \) refers to the approximation using the Trapezoidal Rule. - \( S(n) \) refers to the approximation using Simpson's Rule. - The absolute errors for each method are calculated by comparing the approximation with the exact integral value \( 129,024 \). Simpson's Rule generally provides a more accurate approximation, as evidenced by smaller errors compared to the Trapezoidal Rule, particularly as \( n \) increases.
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