Complete the table for n = 4, 8, 16, and 32 **Title: Approximating Integrals Using the Midpoint and Trapezoid Rules**---**Problem Statement:**Apply the Midpoint and Trapezoid Rules to approximate the following integral, and create 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.\[\int_{1}^{5} (3x^2 - 2x) \, dx = 100\]---**Instructions:**Complete the following table with the approximations and the absolute errors for each method and each value of $ n $.*(Type integers or decimals.)*\[\begin{array}{|c|c|c|c|c|}\hlinen & M(n) & T(n) & \text{Absolute Error in } M(n) & \text{Absolute Error in } T(n) \\\hline4 & \, & \, & \, & \, \\\hline\end{array}\]---**Explanation of Terms:**- **$ M(n) $**: The approximation using the Midpoint Rule with $ n $ subintervals.- **$ T(n) $**: The approximation using the Trapezoid Rule with $ n $ subintervals.- **Absolute Error**: The difference between the exact value and the approximation. This exercise will help in understanding how numerical integration can be used to approximate definite integrals and how the accuracy of the approximation improves with an increasing number of subintervals.

Answered: Apply the Midpoint and Trapezoid Rules…

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

**Title: Approximating Integrals Using the Midpoint and Trapezoid Rules**

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**Problem Statement:**

Apply the Midpoint and Trapezoid Rules to approximate the following integral, and create 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.

\[
\int_{1}^{5} (3x^2 - 2x) \, dx = 100
\]

---

**Instructions:**

Complete the following table with the approximations and the absolute errors for each method and each value of $ n $.

*(Type integers or decimals.)*

\[
\begin{array}{|c|c|c|c|c|}
\hline
n & M(n) & T(n) & \text{Absolute Error in } M(n) & \text{Absolute Error in } T(n) \\
\hline
4 & \, & \, & \, & \, \\
\hline
\end{array}
\]

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**Explanation of Terms:**

- **$ M(n) $**: The approximation using the Midpoint Rule with $ n $ subintervals.
- **$ T(n) $**: The approximation using the Trapezoid Rule with $ n $ subintervals.
- **Absolute Error**: The difference between the exact value and the approximation.

This exercise will help in understanding how numerical integration can be used to approximate definite integrals and how the accuracy of the approximation improves with an increasing number of subintervals.

Expert Solution

Step 1: Analysis and Introduction

Given Information:

$integral subscript 1 superscript 5 open parentheses 3 x squared minus 2 x close parentheses d x equals 100$ .

$n equals 4 comma 8 comma 16.32$ .

To find:

The approximate value of integration by midpoint rule and trapezoid rule.

Also, find the approximate error when compared to exact value.

Concept used:

Midpoint rule: $integral subscript a superscript b f open parentheses x close parentheses d x equals increment x sum from i equals 1 to n of f open parentheses top enclose x subscript i end enclose close parentheses$ , where $top enclose x subscript i end enclose$ is the midpoint of $i to the power of th$ sub-interval and $increment x equals fraction numerator b minus a over denominator n end fraction$ .

Trapezoidal rule: $integral subscript a superscript b f open parentheses x close parentheses d x equals fraction numerator increment x over denominator 2 end fraction open square brackets f open parentheses x subscript 0 close parentheses plus f open parentheses x subscript n close parentheses plus 2 open parentheses f open parentheses x subscript 1 close parentheses plus f open parentheses x subscript 2 close parentheses plus... plus f open parentheses x subscript n minus 1 end subscript close parentheses close parentheses close square brackets$