Apply Simpson's Rule to the following integral. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations. Make a table showing the approximations and errors for n = 4, 8, 16, and 32. The exact value of the integral is given for computing the error. 6 S5e-5 0 Complete the table below. (Type integers or decimals. Round to six decimal places as needed.) T(n) S(n) Absolute Error in T(n) n -5x dx=1-e-301.000000 4 Absolute Error in S(n) ...
Apply Simpson's Rule to the following integral. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations. Make a table showing the approximations and errors for n = 4, 8, 16, and 32. The exact value of the integral is given for computing the error. 6 S5e-5 0 Complete the table below. (Type integers or decimals. Round to six decimal places as needed.) T(n) S(n) Absolute Error in T(n) n -5x dx=1-e-301.000000 4 Absolute Error in S(n) ...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Application of Simpson's Rule for Numerical Integration**
**Objective:**
Apply Simpson's Rule to the following integral to approximate its value. The Trapezoid Rule is first used to generate approximations, which are then utilized to find the Simpson's Rule approximations. A table will be constructed to showcase the approximations and errors for \( n = 4, 8, 16, \) and \( 32 \). The exact value of the integral is provided for computing errors.
**Integral to Evaluate:**
\[
\int_{0}^{6} 5e^{-5x} \, dx = 1 - e^{-30} \approx 1.000000
\]
**Task:**
Complete the table below using the provided approximations.
**Format Instructions:**
(Type integers or decimals. Round to six decimal places as needed.)
| n | \( T(n) \) | \( S(n) \) | Absolute Error in \( T(n) \) | Absolute Error in \( S(n) \) |
|---|------------|------------|-----------------------------|-----------------------------|
| 4 | | | | |
- **Explanation of Columns:**
- **\( n \):** Number of subintervals used in the approximation.
- **\( T(n) \):** Trapezoid Rule approximation for \( n \) subintervals.
- **\( S(n) \):** Simpson's Rule approximation derived from \( T(n) \).
- **Absolute Error in \( T(n) \):** Absolute difference between \( T(n) \) and the exact integral value.
- **Absolute Error in \( S(n) \):** Absolute difference between \( S(n) \) and the exact integral value.
> Note: Ensure all calculations adhere to the specified precision.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F844281c0-c621-44a5-8564-9e61584c69dd%2Fc8f6bdcb-ead4-4165-a08d-f2221abc7997%2Fujcyxl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Application of Simpson's Rule for Numerical Integration**
**Objective:**
Apply Simpson's Rule to the following integral to approximate its value. The Trapezoid Rule is first used to generate approximations, which are then utilized to find the Simpson's Rule approximations. A table will be constructed to showcase the approximations and errors for \( n = 4, 8, 16, \) and \( 32 \). The exact value of the integral is provided for computing errors.
**Integral to Evaluate:**
\[
\int_{0}^{6} 5e^{-5x} \, dx = 1 - e^{-30} \approx 1.000000
\]
**Task:**
Complete the table below using the provided approximations.
**Format Instructions:**
(Type integers or decimals. Round to six decimal places as needed.)
| n | \( T(n) \) | \( S(n) \) | Absolute Error in \( T(n) \) | Absolute Error in \( S(n) \) |
|---|------------|------------|-----------------------------|-----------------------------|
| 4 | | | | |
- **Explanation of Columns:**
- **\( n \):** Number of subintervals used in the approximation.
- **\( T(n) \):** Trapezoid Rule approximation for \( n \) subintervals.
- **\( S(n) \):** Simpson's Rule approximation derived from \( T(n) \).
- **Absolute Error in \( T(n) \):** Absolute difference between \( T(n) \) and the exact integral value.
- **Absolute Error in \( S(n) \):** Absolute difference between \( S(n) \) and the exact integral value.
> Note: Ensure all calculations adhere to the specified precision.
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