Which Riemann sum represents the illustration shown? 3 La 2- 1 3 6 9 3 -Xi • Axi i=1 3 -X¡ • Axi -Xi • Axi O O O WI “WI WI O Σ 1 1/3 Xi
Which Riemann sum represents the illustration shown? 3 La 2- 1 3 6 9 3 -Xi • Axi i=1 3 -X¡ • Axi -Xi • Axi O O O WI “WI WI O Σ 1 1/3 Xi
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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![### Riemann Sums and Area Under a Curve
#### Problem Statement
The given problem asks to identify which Riemann sum represents the illustration provided.
#### Graph Description
The graph illustrated is a piecewise function graph that appears to approximate the area under a curve using three rectangles, summing up the areas to estimate the definite integral.
- The x-axis ranges from 0 to 9.
- The y-axis ranges from 0 to 4.
- The rectangles have a uniform width of 3 units and various heights (1, 2, and 3 units).
#### Choices
The options given for the Riemann sum are:
1. \( \sum_{i=1}^{3} \frac{1}{3} x_i \cdot \Delta x_i \)
2. \( \sum_{i=1}^{3} \frac{2}{3} x_i \cdot \Delta x_i \)
3. \( \sum_{i=1}^{5} \frac{1}{3} x_i \cdot \Delta x_i \)
4. \( \sum_{i=1}^{3} \frac{1}{3} x_i \)
#### Riemann Sum Explanation
A Riemann sum calculates the approximate area under a curve by summing up the areas of rectangles. Each rectangle's area is given by \( f(x_i) \cdot \Delta x \), where \( x_i \) is a sample point in the \( i \)-th subinterval, and \( \Delta x \) is the width of each subinterval.
Here's how to interpret the graph according to the given options:
1. **First Option:**
\[
\sum_{i=1}^{3} \frac{1}{3} x_i \cdot \Delta x_i
\]
Sums the areas using heights scaled by \(\frac{1}{3} x_i\). Here, \( \Delta x_i \) should be the width of each interval, which is 3.
2. **Second Option:**
\[
\sum_{i=1}^{3} \frac{2}{3} x_i \cdot \Delta x_i
\]
Similar to the first but scales by \(\frac{2}{3} x_i\).
3. **Third Option:**
\[
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fb4dfc3-db65-4bb6-9a5b-9427f46c7a3d%2F38af578f-759a-4c76-a17a-6f39fb2060fc%2F77u4dqj_processed.png&w=3840&q=75)
Transcribed Image Text:### Riemann Sums and Area Under a Curve
#### Problem Statement
The given problem asks to identify which Riemann sum represents the illustration provided.
#### Graph Description
The graph illustrated is a piecewise function graph that appears to approximate the area under a curve using three rectangles, summing up the areas to estimate the definite integral.
- The x-axis ranges from 0 to 9.
- The y-axis ranges from 0 to 4.
- The rectangles have a uniform width of 3 units and various heights (1, 2, and 3 units).
#### Choices
The options given for the Riemann sum are:
1. \( \sum_{i=1}^{3} \frac{1}{3} x_i \cdot \Delta x_i \)
2. \( \sum_{i=1}^{3} \frac{2}{3} x_i \cdot \Delta x_i \)
3. \( \sum_{i=1}^{5} \frac{1}{3} x_i \cdot \Delta x_i \)
4. \( \sum_{i=1}^{3} \frac{1}{3} x_i \)
#### Riemann Sum Explanation
A Riemann sum calculates the approximate area under a curve by summing up the areas of rectangles. Each rectangle's area is given by \( f(x_i) \cdot \Delta x \), where \( x_i \) is a sample point in the \( i \)-th subinterval, and \( \Delta x \) is the width of each subinterval.
Here's how to interpret the graph according to the given options:
1. **First Option:**
\[
\sum_{i=1}^{3} \frac{1}{3} x_i \cdot \Delta x_i
\]
Sums the areas using heights scaled by \(\frac{1}{3} x_i\). Here, \( \Delta x_i \) should be the width of each interval, which is 3.
2. **Second Option:**
\[
\sum_{i=1}^{3} \frac{2}{3} x_i \cdot \Delta x_i
\]
Similar to the first but scales by \(\frac{2}{3} x_i\).
3. **Third Option:**
\[
\
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