Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 7x x² + 4 lim f(x) n i = 1 7 n 1 1 ≤ x ≤ 3 X
Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 7x x² + 4 lim f(x) n i = 1 7 n 1 1 ≤ x ≤ 3 X
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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Question
 to find an expression for the area under the graph of \( f \) as a limit. Do not evaluate the limit.
Given function:
\[ f(x) = \frac{7x}{x^2 + 4}, \quad 1 \leq x \leq 3 \]
#### Step-by-Step Solution:
To find the area under the graph of \( f \) from \( x = 1 \) to \( x = 3 \), we use the definition of the definite integral as a limit of Riemann sums:
1. **Divide the Interval:** Divide the interval \([1, 3]\) into \(n\) sub-intervals, each of width \(\Delta x\), where
\[ \Delta x = \frac{3 - 1}{n} = \frac{2}{n}. \]
2. **Define Sample Points:** Let \( x_i^* \) be a sample point in the \(i\)-th sub-interval. Typically, we can use the right end-points or mid-points of each sub-interval.
3. **Sum of Areas:** The Riemann sum for the given function is expressed as:
\[ \sum_{i=1}^{n} f(x_i^*) \Delta x \]
4. **Express \( f(x_i^*) \):** Using \( x_i^* \) in the function \( f(x) \),
\[ f(x_i^*) = \frac{7 x_i^*}{(x_i^*)^2 + 4} \]
5. **Riemann Sum Expression:**
Combine \( f(x_i^*) \) and \(\Delta x \):
\[ \sum_{i=1}^{n} \frac{7 x_i^*}{(x_i^*)^2 + 4} \cdot \frac{2}{n} \]
6. **Limit Definition of the Integral:**
The area under the curve from \( x = 1 \) to \( x = 3 \) is given by the limit of this Riemann sum as \( n \) approaches infinity:
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{7 x_i^*}{(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80fd0a7-9ae8-4bc5-b926-d69bfc10c0d7%2F2c41a08a-efaf-4fa7-bb50-5276156d95fb%2Fg5w54nj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Finding the Area Under the Curve Using the Definition of a Limit
#### Problem Statement:
Use the [Definition](#) to find an expression for the area under the graph of \( f \) as a limit. Do not evaluate the limit.
Given function:
\[ f(x) = \frac{7x}{x^2 + 4}, \quad 1 \leq x \leq 3 \]
#### Step-by-Step Solution:
To find the area under the graph of \( f \) from \( x = 1 \) to \( x = 3 \), we use the definition of the definite integral as a limit of Riemann sums:
1. **Divide the Interval:** Divide the interval \([1, 3]\) into \(n\) sub-intervals, each of width \(\Delta x\), where
\[ \Delta x = \frac{3 - 1}{n} = \frac{2}{n}. \]
2. **Define Sample Points:** Let \( x_i^* \) be a sample point in the \(i\)-th sub-interval. Typically, we can use the right end-points or mid-points of each sub-interval.
3. **Sum of Areas:** The Riemann sum for the given function is expressed as:
\[ \sum_{i=1}^{n} f(x_i^*) \Delta x \]
4. **Express \( f(x_i^*) \):** Using \( x_i^* \) in the function \( f(x) \),
\[ f(x_i^*) = \frac{7 x_i^*}{(x_i^*)^2 + 4} \]
5. **Riemann Sum Expression:**
Combine \( f(x_i^*) \) and \(\Delta x \):
\[ \sum_{i=1}^{n} \frac{7 x_i^*}{(x_i^*)^2 + 4} \cdot \frac{2}{n} \]
6. **Limit Definition of the Integral:**
The area under the curve from \( x = 1 \) to \( x = 3 \) is given by the limit of this Riemann sum as \( n \) approaches infinity:
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{7 x_i^*}{(
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