How large should n be to guarantee that the Trapezoidal Rule approximation to -1 (-24 x² – 10x³ - 10x³ – 24x² - 4x + 4) dx is accurate to within 0.1. n =
How large should n be to guarantee that the Trapezoidal Rule approximation to -1 (-24 x² – 10x³ - 10x³ – 24x² - 4x + 4) dx is accurate to within 0.1. n =
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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![**Educational Content on Numerical Integration Techniques**
---
### Problem Statement:
**How large should \( n \) be to guarantee that the Trapezoidal Rule approximation to**
\[
\int_{-4}^{-1} \left( -x^4 - 10x^3 - 24x^2 - 4x + 4 \right) dx
\]
**is accurate to within 0.1.**
\( n = \) [ Text Box ]
---
**How large should \( n \) be to guarantee that the Simpson's Rule approximation to**
\[
\int_{-4}^{-1} \left( -x^4 - 10x^3 - 24x^2 - 4x + 4 \right) dx
\]
**is accurate to within 0.1.**
\( n = \) [ Text Box ]
---
**Hint:** Remember your answers should be whole numbers, and Simpson's Rule requires even values for \( n \).
---
In this problem, students are asked to determine the number of subintervals \( n \) needed to achieve a specified accuracy when approximating a definite integral using the Trapezoidal Rule and Simpson's Rule. The integral represents a polynomial function that is to be integrated over the interval from \(-4\) to \(-1\). The accuracy criterion specified is 0.1.
For better understanding:
- **Trapezoidal Rule** uses trapezoids to approximate the area under the curve, which requires considering the error bound formula for the Trapezoidal Rule to ensure the accuracy.
- **Simpson's Rule** leverages parabolic segments to approximate the area under the curve. This rule typically offers better accuracy for a given \( n \) compared to the Trapezoidal Rule but also requires \( n \) to be even.
This exercise necessitates understanding the application of numerical methods for integration, specifically focusing on error estimation techniques.
---
*Note:* To insert numerical solutions into the text boxes, students will need to apply error bounds for each method and solve for \( n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc31d976c-acd4-451e-8eb9-fde305025b60%2F46d144a4-84c1-431c-b9ad-5cbe85e7d538%2Fh8q93o_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content on Numerical Integration Techniques**
---
### Problem Statement:
**How large should \( n \) be to guarantee that the Trapezoidal Rule approximation to**
\[
\int_{-4}^{-1} \left( -x^4 - 10x^3 - 24x^2 - 4x + 4 \right) dx
\]
**is accurate to within 0.1.**
\( n = \) [ Text Box ]
---
**How large should \( n \) be to guarantee that the Simpson's Rule approximation to**
\[
\int_{-4}^{-1} \left( -x^4 - 10x^3 - 24x^2 - 4x + 4 \right) dx
\]
**is accurate to within 0.1.**
\( n = \) [ Text Box ]
---
**Hint:** Remember your answers should be whole numbers, and Simpson's Rule requires even values for \( n \).
---
In this problem, students are asked to determine the number of subintervals \( n \) needed to achieve a specified accuracy when approximating a definite integral using the Trapezoidal Rule and Simpson's Rule. The integral represents a polynomial function that is to be integrated over the interval from \(-4\) to \(-1\). The accuracy criterion specified is 0.1.
For better understanding:
- **Trapezoidal Rule** uses trapezoids to approximate the area under the curve, which requires considering the error bound formula for the Trapezoidal Rule to ensure the accuracy.
- **Simpson's Rule** leverages parabolic segments to approximate the area under the curve. This rule typically offers better accuracy for a given \( n \) compared to the Trapezoidal Rule but also requires \( n \) to be even.
This exercise necessitates understanding the application of numerical methods for integration, specifically focusing on error estimation techniques.
---
*Note:* To insert numerical solutions into the text boxes, students will need to apply error bounds for each method and solve for \( n \).
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