Apply Simpson's Rule to the following integral. It is easiest to obtain the Simpson's Rule Make a table showing the approximations and errors for n = 4, 8, 16, and 32. The exact 8 n 0 1125- *** (3x5 - 2x³) dx = 129,024 4 8 16 32 T(n) 149120 134096 130295 129341.9 S(n) 129088 129028 129024.25 Absolute Error in T(n) 20096 5072 1271 317.9 --- ... Absolute Error in S(n) - 64 4 0.25
Apply Simpson's Rule to the following integral. It is easiest to obtain the Simpson's Rule Make a table showing the approximations and errors for n = 4, 8, 16, and 32. The exact 8 n 0 1125- *** (3x5 - 2x³) dx = 129,024 4 8 16 32 T(n) 149120 134096 130295 129341.9 S(n) 129088 129028 129024.25 Absolute Error in T(n) 20096 5072 1271 317.9 --- ... Absolute Error in S(n) - 64 4 0.25
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with n=32
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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