Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers to four decimal places and compare the results with the exact value of the definite integral. [² Trapezoidal Simpson's exact x² dx, n = 4 Need Help? Read It Watch It Master It
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answers to four decimal places and compare the results with the exact value of the definite integral. [² Trapezoidal Simpson's exact x² dx, n = 4 Need Help? Read It Watch It Master It
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
![**Title: Approximating Definite Integrals Using Numerical Methods**
**Objective:**
Learn to use the Trapezoidal Rule and Simpson's Rule to approximate the value of a definite integral. Compare these approximations with the exact value of the definite integral.
**Problem Statement:**
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \( n \). Round your answers to four decimal places and compare the results with the exact value of the definite integral.
\[ \int_{0}^{2} x^2 \, dx, \quad n = 4 \]
**Fields to Fill:**
- **Trapezoidal:** [Enter your calculated value here]
- **Simpson's:** [Enter your calculated value here]
- **Exact:** [Enter the exact value here]
**Need Help?**
- **Read It**
- **Watch It**
- **Master It**
**Instructions:**
1. Calculate the approximate value using the Trapezoidal Rule.
2. Calculate the approximate value using Simpson's Rule.
3. Find the exact value of the integral.
4. Compare and discuss the results. Consider the accuracy and efficiency of each method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb008c437-902e-4304-877f-834a67eaa9dc%2Fef336d78-5c69-4f66-894c-dc3ec9ebdb7c%2Fky7age7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Approximating Definite Integrals Using Numerical Methods**
**Objective:**
Learn to use the Trapezoidal Rule and Simpson's Rule to approximate the value of a definite integral. Compare these approximations with the exact value of the definite integral.
**Problem Statement:**
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \( n \). Round your answers to four decimal places and compare the results with the exact value of the definite integral.
\[ \int_{0}^{2} x^2 \, dx, \quad n = 4 \]
**Fields to Fill:**
- **Trapezoidal:** [Enter your calculated value here]
- **Simpson's:** [Enter your calculated value here]
- **Exact:** [Enter the exact value here]
**Need Help?**
- **Read It**
- **Watch It**
- **Master It**
**Instructions:**
1. Calculate the approximate value using the Trapezoidal Rule.
2. Calculate the approximate value using Simpson's Rule.
3. Find the exact value of the integral.
4. Compare and discuss the results. Consider the accuracy and efficiency of each method.
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