Consider the integral 6 Sod X dx with n = 4. 1 a. Find the trapezoid rule approximations to the integral using n and 2n subintervals. b. Find the Simpson's rule approximation to the integral using 2n subintervals.
Consider the integral 6 Sod X dx with n = 4. 1 a. Find the trapezoid rule approximations to the integral using n and 2n subintervals. b. Find the Simpson's rule approximation to the integral using 2n subintervals.
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 T(8) using the trapezoid rule and T(4) and T(8) using the Simpsons rule
![**Educational Resource on Numerical Integration**
---
**Consider the Integral**
\[
\int_{1}^{e} \frac{6}{x} \, dx \quad \text{with} \quad n = 4.
\]
**Tasks:**
a. Find the trapezoid rule approximations to the integral using \( n \) and \( 2n \) subintervals.
b. Find the Simpson's rule approximation to the integral using \( 2n \) subintervals.
c. Compute the absolute errors in the trapezoid rule and Simpson's rule with \( 2n \) subintervals.
---
**a. What is the trapezoid approximation with \( n \) subintervals?**
\[ T(4) = 6.078234 \]
*(Round to six decimal places as needed.)*
**What is the trapezoid approximation with \( 2n \) subintervals?**
\[ T(8) = \]
*(Round to six decimal places as needed.)*
---
This section is designed to help students understand and apply numerical integration techniques such as the trapezoid rule and Simpson's rule to approximate the value of definite integrals. It provides a practical example and encourages the calculation and comparison of approximations using different numbers of subintervals, emphasizing accuracy and error analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa204fb01-1593-427c-b210-f132b42420bd%2F37b296ea-5252-4e00-952b-d0fcdbf9be76%2F7c6zdo8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Resource on Numerical Integration**
---
**Consider the Integral**
\[
\int_{1}^{e} \frac{6}{x} \, dx \quad \text{with} \quad n = 4.
\]
**Tasks:**
a. Find the trapezoid rule approximations to the integral using \( n \) and \( 2n \) subintervals.
b. Find the Simpson's rule approximation to the integral using \( 2n \) subintervals.
c. Compute the absolute errors in the trapezoid rule and Simpson's rule with \( 2n \) subintervals.
---
**a. What is the trapezoid approximation with \( n \) subintervals?**
\[ T(4) = 6.078234 \]
*(Round to six decimal places as needed.)*
**What is the trapezoid approximation with \( 2n \) subintervals?**
\[ T(8) = \]
*(Round to six decimal places as needed.)*
---
This section is designed to help students understand and apply numerical integration techniques such as the trapezoid rule and Simpson's rule to approximate the value of definite integrals. It provides a practical example and encourages the calculation and comparison of approximations using different numbers of subintervals, emphasizing accuracy and error analysis.
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