n Rn ² = ²/12 (-8 + + ²). n lim Rn n→→∞ = dx.
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
![**Problem Statement:**
Express the limit of \( R_n \) as a definite integral, identifying the correct interval.
**Given Expression:**
\[ R_n = \frac{4}{n} \sum_{i=1}^{n} \left( -8 + i \frac{4}{n} \right) \]
**Objective:**
Find the limit as \( n \to \infty \):
\[ \lim_{n \to \infty} R_n = \int_{[ \, ]}^{[ \, ]} [ \, ] \, dx. \]
**Explanation:**
- The left side of the equation represents the limit of the Riemann sum.
- The right side shows an incomplete definite integral with spaces for limits of integration and the integrand, indicating the need to express the sum as an integral over a specific interval.
**Steps to Solve:**
1. Recognize the Riemann sum structure in the given expression.
2. Identify the function within the sum.
3. Determine the interval over which the integral needs to be computed, based on the given expression.
**Visual Summary:**
There are two components:
1. An equation representing a Riemann sum.
2. The corresponding indefinite integral with placeholders for the interval and function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7dcbfcd-ec40-4419-b8e9-2a693a942e70%2F5bd38754-e858-4464-a953-e604394cb694%2F8czpjnh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Express the limit of \( R_n \) as a definite integral, identifying the correct interval.
**Given Expression:**
\[ R_n = \frac{4}{n} \sum_{i=1}^{n} \left( -8 + i \frac{4}{n} \right) \]
**Objective:**
Find the limit as \( n \to \infty \):
\[ \lim_{n \to \infty} R_n = \int_{[ \, ]}^{[ \, ]} [ \, ] \, dx. \]
**Explanation:**
- The left side of the equation represents the limit of the Riemann sum.
- The right side shows an incomplete definite integral with spaces for limits of integration and the integrand, indicating the need to express the sum as an integral over a specific interval.
**Steps to Solve:**
1. Recognize the Riemann sum structure in the given expression.
2. Identify the function within the sum.
3. Determine the interval over which the integral needs to be computed, based on the given expression.
**Visual Summary:**
There are two components:
1. An equation representing a Riemann sum.
2. The corresponding indefinite integral with placeholders for the interval and function.
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