Math Problem Statement

Homework:HW SECTION 8.8 Question 4, 8.8.45 Part 4 of 5 HW Score: 75%, 6 of 8 points Points: 0 of 1

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question content area top Part 1 Consider the integral Integral from 0 to 1 e Superscript 7 x Baseline dx with nequals20. 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. Question content area bottom Part 1 a. What is the trapezoid approximation with n​ subintervals? Upper T left parenthesis 20 right parenthesisequals    158.113568 ​(Round to six decimal places as​ needed.) Part 2 What is the trapezoid approximation with 2n​ subintervals? Upper T left parenthesis 40 right parenthesisequals    156.918268 ​(Round to six decimal places as​ needed.) Part 3 b. What is the​ Simpson's rule approximation with 2n​ subintervals? Upper S left parenthesis 40 right parenthesisequals    156.519835 ​(Round to six decimal places as​ needed.) Part 4 c. What is the error in the trapezoid rule approximation with 2n​ subintervals?    enter your response here ​(Use scientific notation. Round to three decimal places as needed. Use the multiplication symbol in the math palette as​ needed.) Upper T left parenthesis 20 right parenthesis(Round to six decimal places as needed.)Upper T left parenthesis 40 right parenthesis(Round to six decimal places as needed.)Upper S left parenthesis 40 right parenthesis(Round to six decimal places as needed.)What is the error in the trapezoid rule approximation with 2n subintervals?(Use scientific notation. Round to three decimal places as needed. Use the multiplication symbol in the math palette as needed.) Get more help

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Math Problem Analysis

Mathematical Concepts

Calculus
Numerical Integration
Trapezoid Rule
Simpson's Rule
Error Analysis

Formulas

Trapezoid Rule: T_n = (b-a)/2n * [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]
Simpson's Rule: S_{2n} = (b-a)/6n * [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{2n-1}) + f(x_{2n})]
Exact Value of Integral: ∫_0^1 e^{7x} dx = (e^7 - 1)/7

Theorems

Fundamental Theorem of Calculus
Error Bound Theorem for Numerical Integration

Suitable Grade Level

College Level (Calculus I)

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