? Question With m = 4 and n = 15, approximate the volume under the graph of g(x, y) = (x³y + 1) y² on the rectangle [0, 2] × [0, 3] using the upper right corner of each subrectangle as the sample point. Hint
? Question With m = 4 and n = 15, approximate the volume under the graph of g(x, y) = (x³y + 1) y² on the rectangle [0, 2] × [0, 3] using the upper right corner of each subrectangle as the sample point. Hint
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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![### Problem Statement
With \( m = 4 \) and \( n = 15 \), approximate the volume under the graph of \( g(x, y) = \left(x^3 y + 1\right)^2 \) on the rectangle \([0, 2] \times [0, 3]\) using the upper right corner of each subrectangle as the sample point.
#### Instructions:
1. **Subdivide the Rectangle**: Divide the rectangle \([0, 2] \times [0, 3]\) into \( m \times n \) smaller rectangles.
2. **Sample Points**: Use the upper right corner of each subrectangle as your sample point.
3. **Calculate the Function Value**: For each subrectangle, calculate the value of the function \( g(x, y) \) at the sample point.
4. **Approximate the Volume**: Sum the contributions of all the subrectangles to approximate the volume.
---
#### Steps to Solve the Problem:
1. **Determine the Dimensions of Each Subrectangle**:
- Width (\( \Delta x \)) = \( \frac{2 - 0}{4} = 0.5 \)
- Height (\( \Delta y \)) = \( \frac{3 - 0}{15} = 0.2 \)
2. **Locate the Sample Points**:
- The sample points will be at the upper right corners of each subrectangle. These points are determined by \( \left(x_i, y_j\right) \), where \( x_i \) ranges from \( 0.5 \) to \( 2 \) (incrementing by 0.5) and \( y_j \) ranges from \( 0.2 \) to \( 3 \) (incrementing by 0.2).
3. **Evaluate the Function**:
- At each sample point \( \left(x_i, y_j\right) \), evaluate \( g(x, y) = \left(x^3 y + 1\right)^2 \).
4. **Calculate the Volume Approximation**:
- Multiply the function value at each sample point by the area of the subrectangle (\( \Delta x \times \Delta y = 0.1 \)) and sum these products to approximate the total volume under the graph.
---
### Diagrams and Graphs:
There are](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2Fb2fc5bfb-4cb3-4cd3-82b6-0d0470d1f1ed%2F14d4sj_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
With \( m = 4 \) and \( n = 15 \), approximate the volume under the graph of \( g(x, y) = \left(x^3 y + 1\right)^2 \) on the rectangle \([0, 2] \times [0, 3]\) using the upper right corner of each subrectangle as the sample point.
#### Instructions:
1. **Subdivide the Rectangle**: Divide the rectangle \([0, 2] \times [0, 3]\) into \( m \times n \) smaller rectangles.
2. **Sample Points**: Use the upper right corner of each subrectangle as your sample point.
3. **Calculate the Function Value**: For each subrectangle, calculate the value of the function \( g(x, y) \) at the sample point.
4. **Approximate the Volume**: Sum the contributions of all the subrectangles to approximate the volume.
---
#### Steps to Solve the Problem:
1. **Determine the Dimensions of Each Subrectangle**:
- Width (\( \Delta x \)) = \( \frac{2 - 0}{4} = 0.5 \)
- Height (\( \Delta y \)) = \( \frac{3 - 0}{15} = 0.2 \)
2. **Locate the Sample Points**:
- The sample points will be at the upper right corners of each subrectangle. These points are determined by \( \left(x_i, y_j\right) \), where \( x_i \) ranges from \( 0.5 \) to \( 2 \) (incrementing by 0.5) and \( y_j \) ranges from \( 0.2 \) to \( 3 \) (incrementing by 0.2).
3. **Evaluate the Function**:
- At each sample point \( \left(x_i, y_j\right) \), evaluate \( g(x, y) = \left(x^3 y + 1\right)^2 \).
4. **Calculate the Volume Approximation**:
- Multiply the function value at each sample point by the area of the subrectangle (\( \Delta x \times \Delta y = 0.1 \)) and sum these products to approximate the total volume under the graph.
---
### Diagrams and Graphs:
There are
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