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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
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:** \[ \
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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