Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Converting the Limit to a Definite Integral**
To convert the provided limit to a definite integral, follow these steps:
1. **Identify the function and interval**:
The given limit:
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \left[4(x_i^*)^3 - 3x_i^*\right] \Delta x
\]
represents a Riemann sum for the function \(f(x) = 4x^3 - 3x\).
2. **Determine the interval**:
The interval provided is \([2, 6]\).
Using this information, we can write the limit as a definite integral over the given interval:
\[
\int_{2}^{6} \left( 4x^3 - 3x \right) dx
\]
### Explanation of Graphs/Diagrams:
- **Riemann Sum Representation**:
The summation symbol \(\sum\) indicates that the function \(4(x_i^*)^3 - 3x_i^*\) is summed over \(n\) subintervals, where each subinterval has width \(\Delta x\).
- **Interval Notation**:
The interval \([2, 6]\) is clearly specified to indicate the limits of integration.
This notation and explanation are crucial for understanding the relationship between the limit of a Riemann sum and the integral of a function over a specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fc55a9a-b37f-4ec5-b0ac-e8f1fa01899a%2Ff259a32f-6ef7-4fa6-a7a0-dfaff497f75f%2Fl05y7os_processed.png&w=3840&q=75)
Transcribed Image Text:**Converting the Limit to a Definite Integral**
To convert the provided limit to a definite integral, follow these steps:
1. **Identify the function and interval**:
The given limit:
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \left[4(x_i^*)^3 - 3x_i^*\right] \Delta x
\]
represents a Riemann sum for the function \(f(x) = 4x^3 - 3x\).
2. **Determine the interval**:
The interval provided is \([2, 6]\).
Using this information, we can write the limit as a definite integral over the given interval:
\[
\int_{2}^{6} \left( 4x^3 - 3x \right) dx
\]
### Explanation of Graphs/Diagrams:
- **Riemann Sum Representation**:
The summation symbol \(\sum\) indicates that the function \(4(x_i^*)^3 - 3x_i^*\) is summed over \(n\) subintervals, where each subinterval has width \(\Delta x\).
- **Interval Notation**:
The interval \([2, 6]\) is clearly specified to indicate the limits of integration.
This notation and explanation are crucial for understanding the relationship between the limit of a Riemann sum and the integral of a function over a specified interval.
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