Please help me with this question. **Title: Error Bound in Approximation of Definite Integrals****Objective:**Find a bound on the error in approximating the definite integral using the specified numerical methods.**Problem Statement:**Evaluate the error bounds for the integral:\[\int_{0}^{5} 2e^{-x} \, dx\]using the following methods. Round your answers to five decimal places.**Methods:**(a) **The Trapezoidal Rule** with $ n = 8 $(b) **Simpson’s Rule** with $ n $ intervals**Instructions:**1. Apply the Trapezoidal Rule with $ n = 8 $ subintervals to approximate the integral and find the error bound.2. Use Simpson’s Rule with an appropriate number of subintervals, calculating the error bound accordingly.**Note:**- Ensure to round your error bound calculations to five decimal places.

Answered: Find a bound on the error in…

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 me with this question.

**Title: Error Bound in Approximation of Definite Integrals**

**Objective:**
Find a bound on the error in approximating the definite integral using the specified numerical methods.

**Problem Statement:**

Evaluate the error bounds for the integral:

\[
\int_{0}^{5} 2e^{-x} \, dx
\]

using the following methods. Round your answers to five decimal places.

**Methods:**

(a) **The Trapezoidal Rule** with $ n = 8 $

(b) **Simpson’s Rule** with $ n $ intervals

**Instructions:**

1. Apply the Trapezoidal Rule with $ n = 8 $ subintervals to approximate the integral and find the error bound.

2. Use Simpson’s Rule with an appropriate number of subintervals, calculating the error bound accordingly.

**Note:**
- Ensure to round your error bound calculations to five decimal places.

Expert Solution

Step 1: Description of given data

We need to use error bound for trapezoidal and Simpson rule

For trapezoidal rule $open vertical bar E subscript T close vertical bar less or equal than fraction numerator K left parenthesis b minus a right parenthesis cubed over denominator 12 n squared end fraction space$ where k is maximum value of $open vertical bar f apostrophe apostrophe left parenthesis x right parenthesis close vertical bar$ on interval of $open square brackets a comma b close square brackets$ .

For Simpson rule $open vertical bar E subscript S close vertical bar less or equal than fraction numerator M left parenthesis b minus a right parenthesis to the power of 5 over denominator 180 n to the power of 4 end fraction space$ where M is maximum value of $open vertical bar f to the power of 4 left parenthesis x right parenthesis close vertical bar$ on interval of $open square brackets a comma b close square brackets$ .