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S(8) round to 6 decimal places as needed

S(8) round to 6 decimal places as needed

Advanced Engineering Mathematics
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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S(8) round to 6 decimal places as needed
**Evaluate the Integral Using Numerical Methods** Consider the integral \( \int_{1}^{e} \frac{6}{x} \, dx \) with \( n = 4 \). **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. --- **Trapezoid Rule Approximations:** - \( T(4) = 6.078234 \) (Round to six decimal places as needed.) - What is the trapezoid approximation with \( 2n \) subintervals? - \( T(8) = 6.019843 \) (Round to six decimal places as needed.) **Simpson's Rule Approximation:** - What is the Simpson's rule approximation with \( 2n \) subintervals? - \( S(8) = \_\_\_ \) (Round to six decimal places as needed.) --- This exercise demonstrates the application of numerical approximation methods to evaluate definite integrals, providing insight into the accuracy and efficiency of different techniques like the trapezoid rule and Simpson's rule.
Transcribed Image Text:**Evaluate the Integral Using Numerical Methods** Consider the integral \( \int_{1}^{e} \frac{6}{x} \, dx \) with \( n = 4 \). **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. --- **Trapezoid Rule Approximations:** - \( T(4) = 6.078234 \) (Round to six decimal places as needed.) - What is the trapezoid approximation with \( 2n \) subintervals? - \( T(8) = 6.019843 \) (Round to six decimal places as needed.) **Simpson's Rule Approximation:** - What is the Simpson's rule approximation with \( 2n \) subintervals? - \( S(8) = \_\_\_ \) (Round to six decimal places as needed.) --- This exercise demonstrates the application of numerical approximation methods to evaluate definite integrals, providing insight into the accuracy and efficiency of different techniques like the trapezoid rule and Simpson's rule.
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