Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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8.8. 2
![**Evaluating Definite Integrals Using the Midpoint Rule**
### Problem Statement:
Compute the following estimate of \( \int_{0}^{8} f(x) \, dx \) using the graph in the figure.
M(4)
### Figure Description:
The figure shows the graph of the function \( y = f(x) \), plotted between \( x = 0 \) and \( x = 10 \). The graph has a curvy shape passing through several points on the grid. Key points where the function is evaluated are visible on the plot.
### Estimation Using the Midpoint Rule:
The Midpoint Rule for \( n \) subintervals is given by:
\[ M(n) = \Delta x \left[ f\left(m_1\right) + f\left(m_2\right) + \ldots + f\left(m_n\right) \right] \]
where \( \Delta x \) is the width of each subinterval, and \( m_i \) represents the midpoint of each subinterval.
For this problem, the interval is from 0 to 8, and we need to estimate using 4 subintervals (\( n = 4 \)).
### Calculation:
1. Determine the width of each subinterval, \( \Delta x \):
\[ \Delta x = \frac{8 - 0}{4} = 2 \]
2. Identify the midpoints of each subinterval:
- Subinterval 1: [0, 2], Midpoint at \( x = 1 \)
- Subinterval 2: [2, 4], Midpoint at \( x = 3 \)
- Subinterval 3: [4, 6], Midpoint at \( x = 5 \)
- Subinterval 4: [6, 8], Midpoint at \( x = 7 \)
3. Evaluate the function at each midpoint:
- \( f(1) \approx 2 \) (based on the graph)
- \( f(3) \approx 6 \)
- \( f(5) \approx 4 \)
- \( f(7) \approx 8 \)
4. Apply the Midpoint Rule formula:
\[ M(4) = 2 \left[ f(1) + f(3) + f(5) + f(7) \right] \]
\[ M(4](https://content.bartleby.com/qna-images/question/06c8c348-2e2e-4131-bc9d-de4a7282c792/786a2bed-2fdd-4231-832a-c61ca449fd7a/czpyyt9.jpeg)
Transcribed Image Text:**Evaluating Definite Integrals Using the Midpoint Rule**
### Problem Statement:
Compute the following estimate of \( \int_{0}^{8} f(x) \, dx \) using the graph in the figure.
M(4)
### Figure Description:
The figure shows the graph of the function \( y = f(x) \), plotted between \( x = 0 \) and \( x = 10 \). The graph has a curvy shape passing through several points on the grid. Key points where the function is evaluated are visible on the plot.
### Estimation Using the Midpoint Rule:
The Midpoint Rule for \( n \) subintervals is given by:
\[ M(n) = \Delta x \left[ f\left(m_1\right) + f\left(m_2\right) + \ldots + f\left(m_n\right) \right] \]
where \( \Delta x \) is the width of each subinterval, and \( m_i \) represents the midpoint of each subinterval.
For this problem, the interval is from 0 to 8, and we need to estimate using 4 subintervals (\( n = 4 \)).
### Calculation:
1. Determine the width of each subinterval, \( \Delta x \):
\[ \Delta x = \frac{8 - 0}{4} = 2 \]
2. Identify the midpoints of each subinterval:
- Subinterval 1: [0, 2], Midpoint at \( x = 1 \)
- Subinterval 2: [2, 4], Midpoint at \( x = 3 \)
- Subinterval 3: [4, 6], Midpoint at \( x = 5 \)
- Subinterval 4: [6, 8], Midpoint at \( x = 7 \)
3. Evaluate the function at each midpoint:
- \( f(1) \approx 2 \) (based on the graph)
- \( f(3) \approx 6 \)
- \( f(5) \approx 4 \)
- \( f(7) \approx 8 \)
4. Apply the Midpoint Rule formula:
\[ M(4) = 2 \left[ f(1) + f(3) + f(5) + f(7) \right] \]
\[ M(4
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