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 far 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 far 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
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Question
Complete the table for n = 4, 8, 16, and 32
![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.
\[
\int_0^8 e^{-6x} \, dx = 1 - e^{-48} \approx 1.000000
\]
---
Complete the table below.
(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:
- **T(n)**: Represents the approximation of the integral using the Trapezoid Rule.
- **S(n)**: Represents the approximation of the integral using Simpson's Rule.
- **Absolute Error in T(n)**: The absolute difference between the exact value and the Trapezoid Rule approximation.
- **Absolute Error in S(n)**: The absolute difference between the exact value and the Simpson's Rule approximation.
You will need to calculate these values using the given integral and round them to six decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1913c9c-da59-4e5e-af1e-78b6351c7d9b%2F55ceb4d6-3287-403f-8d59-3528dec0062a%2F19c7ijk_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
\[
\int_0^8 e^{-6x} \, dx = 1 - e^{-48} \approx 1.000000
\]
---
Complete the table below.
(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:
- **T(n)**: Represents the approximation of the integral using the Trapezoid Rule.
- **S(n)**: Represents the approximation of the integral using Simpson's Rule.
- **Absolute Error in T(n)**: The absolute difference between the exact value and the Trapezoid Rule approximation.
- **Absolute Error in S(n)**: The absolute difference between the exact value and the Simpson's Rule approximation.
You will need to calculate these values using the given integral and round them to six decimal places.
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