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What is the Riemann sum that represents the illustration shown? The function is f(x)=√√x. 3 i=1 3 Σ O √xi √xi• Axi

What is the Riemann sum that represents the illustration shown? The function is f(x)=√√x. 3 i=1 3 Σ O √xi √xi• Axi

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 in Calculus - Understanding the Illustration** **Question:** What is the Riemann sum that represents the illustration shown? The function is \( f(x) = \sqrt{x} \). **Illustration Explanation:** The illustration depicts a function, \( f(x) = \sqrt{x} \), with a series of rectangles under the curve. This is a visual representation often used to explain the concept of Riemann sums in calculus. The rectangles approximate the area under the curve, which will aid in understanding integral calculus. **Options for the Riemann Sum:** 1. \( \sum_{i=1}^{3} \sqrt{x_i} \) - This option suggests summing the square root of \( x_i \) from \( i = 1 \) to \( i = 3 \). 2. \( \sum_{i=1}^{3} \sqrt{x_i} \cdot \Delta x_i \) - This option implies summing the product of \( \sqrt{x_i} \) and \( \Delta x_i \), with \( i \) ranging from 1 to 3. 3. \( \sum_{i=1}^{5} \sqrt{x_i} \) - Here, the suggestion is to sum the square root of \( x_i \) from \( i = 1 \) to \( i = 5 \). 4. \( \sum_{i=1}^{5} \sqrt{x_i} \cdot \Delta x_i \) - This option involves summing the product of \( \sqrt{x_i} \) and \( \Delta x_i \), with \( i \) ranging from 1 to 5. Given that the area is approximated by rectangles, the correct Riemann sum should include the width of the rectangles, \( \Delta x_i \). The appropriate choice is the second option, \( \sum_{i=1}^{3} \sqrt{x_i} \cdot \Delta x_i \), as it accurately reflects the sum of the areas of these rectangles under the curve \( f(x) = \sqrt{x} \).
Transcribed Image Text:**Riemann Sums in Calculus - Understanding the Illustration** **Question:** What is the Riemann sum that represents the illustration shown? The function is \( f(x) = \sqrt{x} \). **Illustration Explanation:** The illustration depicts a function, \( f(x) = \sqrt{x} \), with a series of rectangles under the curve. This is a visual representation often used to explain the concept of Riemann sums in calculus. The rectangles approximate the area under the curve, which will aid in understanding integral calculus. **Options for the Riemann Sum:** 1. \( \sum_{i=1}^{3} \sqrt{x_i} \) - This option suggests summing the square root of \( x_i \) from \( i = 1 \) to \( i = 3 \). 2. \( \sum_{i=1}^{3} \sqrt{x_i} \cdot \Delta x_i \) - This option implies summing the product of \( \sqrt{x_i} \) and \( \Delta x_i \), with \( i \) ranging from 1 to 3. 3. \( \sum_{i=1}^{5} \sqrt{x_i} \) - Here, the suggestion is to sum the square root of \( x_i \) from \( i = 1 \) to \( i = 5 \). 4. \( \sum_{i=1}^{5} \sqrt{x_i} \cdot \Delta x_i \) - This option involves summing the product of \( \sqrt{x_i} \) and \( \Delta x_i \), with \( i \) ranging from 1 to 5. Given that the area is approximated by rectangles, the correct Riemann sum should include the width of the rectangles, \( \Delta x_i \). The appropriate choice is the second option, \( \sum_{i=1}^{3} \sqrt{x_i} \cdot \Delta x_i \), as it accurately reflects the sum of the areas of these rectangles under the curve \( f(x) = \sqrt{x} \).
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