Express the limit as a definite integral. lim (c-2ck)Axk, where P is a partition of [ - 3,9] ||P||→0 K=1 n 9 İCC 4 The definite integral is as follows.
Express the limit as a definite integral. lim (c-2ck)Axk, where P is a partition of [ - 3,9] ||P||→0 K=1 n 9 İCC 4 The definite integral is as follows.
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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![**Transcription for Educational Website:**
### Express the Limit as a Definite Integral
\[ \lim_{\lVert P \rVert \to 0} \sum_{k=1}^{n} (c_k^4 - 2c_k) \Delta x_k, \text{ where P is a partition of } [-3, 9] \]
**Explanation:**
This expression represents the limit of a Riemann sum, which is a method for approximating the total area under a curve (the integral). As the partition P becomes finer (i.e., the norm of the partition \(\lVert P \rVert\) approaches 0), the Riemann sum converges to the exact value of the definite integral over the interval \([-3, 9]\).
---
### The Definite Integral is as follows:
\[ \int_{-3}^{9} \]
**Diagram Explanation:**
- The integral sign \(\int\) indicates the operation of integration.
- The limits of integration are from \(-3\) to \(9\), representing the interval over which the function is integrated.
- The integrand, which would be specified in the middle (represented here by a placeholder), is the function resulting from the limit of the Riemann sum: \((c^4 - 2c)\) based on the context of the problem.
This tool can compute the definite integral of a function over a specified interval, contributing to understanding the area under the curve and solving broader calculus problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93ade889-6aa6-440d-b6d3-23ce5a98ca6e%2F3ed4328a-a494-4534-9d0a-36b8048e9ba9%2F43tsaj_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
### Express the Limit as a Definite Integral
\[ \lim_{\lVert P \rVert \to 0} \sum_{k=1}^{n} (c_k^4 - 2c_k) \Delta x_k, \text{ where P is a partition of } [-3, 9] \]
**Explanation:**
This expression represents the limit of a Riemann sum, which is a method for approximating the total area under a curve (the integral). As the partition P becomes finer (i.e., the norm of the partition \(\lVert P \rVert\) approaches 0), the Riemann sum converges to the exact value of the definite integral over the interval \([-3, 9]\).
---
### The Definite Integral is as follows:
\[ \int_{-3}^{9} \]
**Diagram Explanation:**
- The integral sign \(\int\) indicates the operation of integration.
- The limits of integration are from \(-3\) to \(9\), representing the interval over which the function is integrated.
- The integrand, which would be specified in the middle (represented here by a placeholder), is the function resulting from the limit of the Riemann sum: \((c^4 - 2c)\) based on the context of the problem.
This tool can compute the definite integral of a function over a specified interval, contributing to understanding the area under the curve and solving broader calculus problems.
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