Compute the Riemann sum for the given function and region with partition of size n on the x axis and size m on the y-axis. f(x) = xy, 8sxs 14, n= 3, 2sys6 m=2. using upper right corner evaluation point for each subrectangle.
Compute the Riemann sum for the given function and region with partition of size n on the x axis and size m on the y-axis. f(x) = xy, 8sxs 14, n= 3, 2sys6 m=2. using upper right corner evaluation point for each subrectangle.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Riemann Sum Calculation for a Given Function
This section will guide you through the process of computing the Riemann sum for a specific function over a rectangular region. The instructions below pertain to a function \( f(x) = xy \) within the given bounds on the x-axis and y-axis, with specified partition sizes.
#### Problem Description:
Given the function:
\[ f(x) = xy \]
The region is defined by:
\[ 8 \leq x \leq 14, \quad n = 3 \]
\[ 2 \leq y \leq 6, \quad m = 2 \]
The task is to compute the Riemann sum using the upper right corner evaluation point for each subrectangle.
#### Steps to Compute the Riemann Sum:
1. **Determine the width of each partition on the x-axis (Δx)**:
\[
\Delta x = \frac{14 - 8}{3} = 2
\]
2. **Determine the width of each partition on the y-axis (Δy)**:
\[
\Delta y = \frac{6 - 2}{2} = 2
\]
3. **List the upper right corner points for evaluation**:
- For the subrectangle (i=1, j=1): \( (x, y) = (10, 4) \)
- For the subrectangle (i=2, j=1): \( (x, y) = (12, 4) \)
- For the subrectangle (i=3, j=1): \( (x, y) = (14, 4) \)
- For the subrectangle (i=1, j=2): \( (x, y) = (10, 6) \)
- For the subrectangle (i=2, j=2): \( (x, y) = (12, 6) \)
- For the subrectangle (i=3, j=2): \( (x, y) = (14, 6) \)
4. **Compute the function value at each evaluation point and multiply by the area of each subrectangle (Δx * Δy)**:
- \( f(10, 4) * 4 = 10 * 4 * 4 = 160 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee1f5d00-2b06-4d9b-996b-2e4c16434e95%2F65a1bdbf-d0c0-4a1e-a352-7d4792af0d6f%2Fnrvgb8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Riemann Sum Calculation for a Given Function
This section will guide you through the process of computing the Riemann sum for a specific function over a rectangular region. The instructions below pertain to a function \( f(x) = xy \) within the given bounds on the x-axis and y-axis, with specified partition sizes.
#### Problem Description:
Given the function:
\[ f(x) = xy \]
The region is defined by:
\[ 8 \leq x \leq 14, \quad n = 3 \]
\[ 2 \leq y \leq 6, \quad m = 2 \]
The task is to compute the Riemann sum using the upper right corner evaluation point for each subrectangle.
#### Steps to Compute the Riemann Sum:
1. **Determine the width of each partition on the x-axis (Δx)**:
\[
\Delta x = \frac{14 - 8}{3} = 2
\]
2. **Determine the width of each partition on the y-axis (Δy)**:
\[
\Delta y = \frac{6 - 2}{2} = 2
\]
3. **List the upper right corner points for evaluation**:
- For the subrectangle (i=1, j=1): \( (x, y) = (10, 4) \)
- For the subrectangle (i=2, j=1): \( (x, y) = (12, 4) \)
- For the subrectangle (i=3, j=1): \( (x, y) = (14, 4) \)
- For the subrectangle (i=1, j=2): \( (x, y) = (10, 6) \)
- For the subrectangle (i=2, j=2): \( (x, y) = (12, 6) \)
- For the subrectangle (i=3, j=2): \( (x, y) = (14, 6) \)
4. **Compute the function value at each evaluation point and multiply by the area of each subrectangle (Δx * Δy)**:
- \( f(10, 4) * 4 = 10 * 4 * 4 = 160 \)
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