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. 8 for the de 6e-6x dx=1-e-48≈ 1.000000 0 Complete the table below. (Type integers or decimals. Round to six decimal places as needed.) Absolute Error T(n) S(n) in T(n) n 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. 8 for the de 6e-6x dx=1-e-48≈ 1.000000 0 Complete the table below. (Type integers or decimals. Round to six decimal places as needed.) Absolute Error T(n) S(n) in T(n) n 4 Absolute Error in S(n)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
complete the table for n = 4, 8, 16, and 32. round to six places after the decimal.
![### Applying Simpson's Rule to Evaluate an Integral
To evaluate the following integral, it is effective to apply Simpson's Rule. Initially, Simpson's Rule approximations can be derived from the Trapezoid Rule approximations. A table will be created to display these approximations and the corresponding errors for \( n = 4, 8, 16, \) and \( 32 \). The exact value of the integral, which is crucial for calculating the error, is given below:
\[
\int_{0}^{8} 6e^{-6x} \, dx = 1 - e^{-48} \approx 1.000000
\]
### Task: Complete the Table Below
You are required to complete the following table by entering integers or decimals and rounding them to six decimal places where necessary.
| \( n \) | \( T(n) \) | \( S(n) \) | Absolute Error in \( T(n) \) | Absolute Error in \( S(n) \) |
|---------|------------|------------|------------------------------|------------------------------|
| 4 | | | | |
Here, \( T(n) \) represents the Trapezoid Rule approximation, while \( S(n) \) signifies the Simpson's Rule approximation. The absolute errors are calculated based on the difference between the exact value of the integral and each of the rule's approximations.
### Instructions
- Use the indicated methods to compute values for \( T(n) \) and \( S(n) \).
- Calculate the absolute errors by comparing the approximations with the exact integral value.
- Enter the computed values in the given table accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1913c9c-da59-4e5e-af1e-78b6351c7d9b%2F142064d1-0083-43d6-800f-a065a061b718%2F83vzy95_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Applying Simpson's Rule to Evaluate an Integral
To evaluate the following integral, it is effective to apply Simpson's Rule. Initially, Simpson's Rule approximations can be derived from the Trapezoid Rule approximations. A table will be created to display these approximations and the corresponding errors for \( n = 4, 8, 16, \) and \( 32 \). The exact value of the integral, which is crucial for calculating the error, is given below:
\[
\int_{0}^{8} 6e^{-6x} \, dx = 1 - e^{-48} \approx 1.000000
\]
### Task: Complete the Table Below
You are required to complete the following table by entering integers or decimals and rounding them to six decimal places where necessary.
| \( n \) | \( T(n) \) | \( S(n) \) | Absolute Error in \( T(n) \) | Absolute Error in \( S(n) \) |
|---------|------------|------------|------------------------------|------------------------------|
| 4 | | | | |
Here, \( T(n) \) represents the Trapezoid Rule approximation, while \( S(n) \) signifies the Simpson's Rule approximation. The absolute errors are calculated based on the difference between the exact value of the integral and each of the rule's approximations.
### Instructions
- Use the indicated methods to compute values for \( T(n) \) and \( S(n) \).
- Calculate the absolute errors by comparing the approximations with the exact integral value.
- Enter the computed values in the given table accordingly.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 6 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
