Use the given transformation to evaluate the integral. SS. (10x+15y) dA, where R is the parallelogram with vertices (-3, 12), (3,-12), (5, -10), and (-1, 14); x = = /²/ (u + v), y = // (v − 4u)
Use the given transformation to evaluate the integral. SS. (10x+15y) dA, where R is the parallelogram with vertices (-3, 12), (3,-12), (5, -10), and (-1, 14); x = = /²/ (u + v), y = // (v − 4u)
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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![The problem presented involves evaluating an integral over a specified region \( R \), which is a parallelogram defined by specific vertices. The given transformation allows the expression of the integral in terms of new variables \( u \) and \( v \). Below is a detailed transcription and explanation:
---
**Problem Statement:**
Use the given transformation to evaluate the integral:
\[
\iint_R (10x + 15y) \, dA,
\]
where \( R \) is the parallelogram with vertices \((-3, 12)\), \((3, -12)\), \((5, -10)\), and \((-1, 14)\).
**Transformation Equations:**
- \( x = \frac{1}{5}(u + v) \)
- \( y = \frac{1}{5}(v - 4u) \)
**Student Input:**
- Calculation Result: 375
- Status: Incorrect (indicated by a red 'X')
**Support Options:**
- Need Help? [Read it]
**Additional Section:**
- [Show My Work (Optional)]
**Weather and System Information:**
- Weather: 40°F, Sunny
- Timestamp: 4:06 (11/4/2023)
---
**Explanation of the Problem:**
This mathematical problem involves calculating a double integral over a region \( R \), which is defined as a parallelogram. The vertices of \( R \) are provided, and they allow for understanding the shape and extent of the region.
A transformation involving the variables \( u \) and \( v \) is provided to facilitate the integration by converting it to a potentially simpler form. The transformation given relates the original variables \( x \) and \( y \) of the integral to the new variables \( u \) and \( v \).
The integral function \( 10x + 15y \) suggests a linear density function in the plane of the parallelogram, and applying the transformation will allow integration in the \( uv \)-plane.
The student's solution entered a result of 375, which was marked incorrect, suggesting a need for revision or further understanding. An option for assistance labeled "Need Help? Read it" is available, indicating resources for learning or assistance.
The "Show My Work" section is optional and might be used for students to input their step-by-step calculations for self-review or grading purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6147cd51-fae9-4b86-867f-e7a7fe5750a1%2Fa17772ce-f5ac-40c2-af54-67169d0cc601%2Ftkrqnp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem presented involves evaluating an integral over a specified region \( R \), which is a parallelogram defined by specific vertices. The given transformation allows the expression of the integral in terms of new variables \( u \) and \( v \). Below is a detailed transcription and explanation:
---
**Problem Statement:**
Use the given transformation to evaluate the integral:
\[
\iint_R (10x + 15y) \, dA,
\]
where \( R \) is the parallelogram with vertices \((-3, 12)\), \((3, -12)\), \((5, -10)\), and \((-1, 14)\).
**Transformation Equations:**
- \( x = \frac{1}{5}(u + v) \)
- \( y = \frac{1}{5}(v - 4u) \)
**Student Input:**
- Calculation Result: 375
- Status: Incorrect (indicated by a red 'X')
**Support Options:**
- Need Help? [Read it]
**Additional Section:**
- [Show My Work (Optional)]
**Weather and System Information:**
- Weather: 40°F, Sunny
- Timestamp: 4:06 (11/4/2023)
---
**Explanation of the Problem:**
This mathematical problem involves calculating a double integral over a region \( R \), which is defined as a parallelogram. The vertices of \( R \) are provided, and they allow for understanding the shape and extent of the region.
A transformation involving the variables \( u \) and \( v \) is provided to facilitate the integration by converting it to a potentially simpler form. The transformation given relates the original variables \( x \) and \( y \) of the integral to the new variables \( u \) and \( v \).
The integral function \( 10x + 15y \) suggests a linear density function in the plane of the parallelogram, and applying the transformation will allow integration in the \( uv \)-plane.
The student's solution entered a result of 375, which was marked incorrect, suggesting a need for revision or further understanding. An option for assistance labeled "Need Help? Read it" is available, indicating resources for learning or assistance.
The "Show My Work" section is optional and might be used for students to input their step-by-step calculations for self-review or grading purposes.
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