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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![**Title: Calculating the Area Under the Curve Using Riemann Sums**
**Objective:**
Learn how to use Riemann Sums to determine the area under a curve for the specific function.
**Problem Statement:**
Use \( R_6 \) to determine the area under the curve for \( f(x) = \frac{1}{x(x-1)} \) on the interval \([2, 5]\).
**Instructions:**
1. **Understand the function and interval:**
- The function provided is \( f(x) = \frac{1}{x(x-1)} \).
- The interval on which we need to calculate the area is \([2, 5]\).
2. **Define Riemann Sum:**
- Riemann Sums provide an approximation of the area under a curve over a specified interval.
- \( R_n \) represents a Riemann Sum with \( n \) subintervals.
3. **Steps to Calculate \( R_6 \):**
- **Step 1:** Determine the width of each subinterval (\( \Delta x \)):
\[
\Delta x = \frac{b-a}{n} = \frac{5-2}{6} = 0.5
\]
- **Step 2:** Identify the endpoints of each subinterval:
\[
a = 2, \quad a+\Delta x = 2.5, \quad a + 2\Delta x = 3, \quad a + 3\Delta x = 3.5, \quad a + 4\Delta x = 4, \quad a + 5\Delta x = 4.5, \quad b = 5
\]
- **Step 3:** Evaluate the function \( f(x) \) at each right endpoint. These points \( x_i \) are \( 2.5, 3, 3.5, 4, 4.5, 5 \).
- **Step 4:** Calculate the sum:
\[
R_6 = \sum_{i=1}^{6} f(x_i) \Delta x
\]
Substitute \( x_i \) values and \( f(x_i) \) into the sum and calculate each term.
4. **Conclusion:**
The result from](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cd02c50-b0b4-4862-9c83-ad8fe712a454%2Ffae8f80d-e6e1-4799-aabc-52f608a6efbb%2Frdttdao_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Area Under the Curve Using Riemann Sums**
**Objective:**
Learn how to use Riemann Sums to determine the area under a curve for the specific function.
**Problem Statement:**
Use \( R_6 \) to determine the area under the curve for \( f(x) = \frac{1}{x(x-1)} \) on the interval \([2, 5]\).
**Instructions:**
1. **Understand the function and interval:**
- The function provided is \( f(x) = \frac{1}{x(x-1)} \).
- The interval on which we need to calculate the area is \([2, 5]\).
2. **Define Riemann Sum:**
- Riemann Sums provide an approximation of the area under a curve over a specified interval.
- \( R_n \) represents a Riemann Sum with \( n \) subintervals.
3. **Steps to Calculate \( R_6 \):**
- **Step 1:** Determine the width of each subinterval (\( \Delta x \)):
\[
\Delta x = \frac{b-a}{n} = \frac{5-2}{6} = 0.5
\]
- **Step 2:** Identify the endpoints of each subinterval:
\[
a = 2, \quad a+\Delta x = 2.5, \quad a + 2\Delta x = 3, \quad a + 3\Delta x = 3.5, \quad a + 4\Delta x = 4, \quad a + 5\Delta x = 4.5, \quad b = 5
\]
- **Step 3:** Evaluate the function \( f(x) \) at each right endpoint. These points \( x_i \) are \( 2.5, 3, 3.5, 4, 4.5, 5 \).
- **Step 4:** Calculate the sum:
\[
R_6 = \sum_{i=1}^{6} f(x_i) \Delta x
\]
Substitute \( x_i \) values and \( f(x_i) \) into the sum and calculate each term.
4. **Conclusion:**
The result from
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