Find the approximation S, to the integral xe dx for n = 6 and then compute the corresponding error Es Select the correct answer. The choices are rounded to 6 decimal places. -0.01707 -0.01207 -0.01327 -0.001509 0.08793 0.00893
Find the approximation S, to the integral xe dx for n = 6 and then compute the corresponding error Es Select the correct answer. The choices are rounded to 6 decimal places. -0.01707 -0.01207 -0.01327 -0.001509 0.08793 0.00893
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
![**Approximation and Error Calculation for Integral**
To solve the integral \(\int_{-1}^{2} xe^{x} dx\) for \(n = 6\), follow the steps below to approximate the value and compute the corresponding error \( E_{s} \):
**Problem Statement:**
Find the approximation \( S_{n} \) to the integral \(\int_{-1}^{2} xe^{x} dx\) for \(n = 6\) and then compute the corresponding error \( E_{s} \).
**Options:**
Select the correct answer. The choices are rounded to 6 decimal places.
- \(\circ\) -0.01707
- \(\circ\) -0.01207
- \(\circ\) -0.01327
- \(\circ\) -0.001509
- \(\circ\) 0.08793
- \(\circ\) 0.00893
**Explanation:**
To approach this problem, we will generally use numerical methods such as the Trapezoidal Rule or Simpson's Rule if specific steps to compute \( S_{n} \) and \( E_{s} \) are not provided. These numerical methods help in approximating definite integrals by subdividing the domain into smaller sections and summing up the area under the curve for each section.
**Trapezoidal Rule Method Example:**
1. Divide the interval \([-1, 2]\) into \(n = 6\) subintervals.
2. Calculate the width of each subinterval \(\Delta x = \frac{2 - (-1)}{6} = 0.5\).
3. Evaluate the function \(xe^{x}\) at each subinterval endpoint.
4. Apply the Trapezoidal Rule formula to get the approximation.
**Simpson's Rule Method Example:**
1. Similar to the Trapezoidal Rule, divide the interval \([-1, 2]\) into \(n = 6\) subintervals.
2. Simpson's Rule requires an even number of intervals.
3. Apply the Simpson's Rule formula to get the approximation.
The correct answer among the choices will match the result of your calculations and should be rounded to 6 decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2Fd98a2960-3897-4c0c-a328-eadb995c5f4a%2Fgmgtxys_processed.png&w=3840&q=75)
Transcribed Image Text:**Approximation and Error Calculation for Integral**
To solve the integral \(\int_{-1}^{2} xe^{x} dx\) for \(n = 6\), follow the steps below to approximate the value and compute the corresponding error \( E_{s} \):
**Problem Statement:**
Find the approximation \( S_{n} \) to the integral \(\int_{-1}^{2} xe^{x} dx\) for \(n = 6\) and then compute the corresponding error \( E_{s} \).
**Options:**
Select the correct answer. The choices are rounded to 6 decimal places.
- \(\circ\) -0.01707
- \(\circ\) -0.01207
- \(\circ\) -0.01327
- \(\circ\) -0.001509
- \(\circ\) 0.08793
- \(\circ\) 0.00893
**Explanation:**
To approach this problem, we will generally use numerical methods such as the Trapezoidal Rule or Simpson's Rule if specific steps to compute \( S_{n} \) and \( E_{s} \) are not provided. These numerical methods help in approximating definite integrals by subdividing the domain into smaller sections and summing up the area under the curve for each section.
**Trapezoidal Rule Method Example:**
1. Divide the interval \([-1, 2]\) into \(n = 6\) subintervals.
2. Calculate the width of each subinterval \(\Delta x = \frac{2 - (-1)}{6} = 0.5\).
3. Evaluate the function \(xe^{x}\) at each subinterval endpoint.
4. Apply the Trapezoidal Rule formula to get the approximation.
**Simpson's Rule Method Example:**
1. Similar to the Trapezoidal Rule, divide the interval \([-1, 2]\) into \(n = 6\) subintervals.
2. Simpson's Rule requires an even number of intervals.
3. Apply the Simpson's Rule formula to get the approximation.
The correct answer among the choices will match the result of your calculations and should be rounded to 6 decimal places.
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