Find (4x + 2y)dA where R is the parallelogram with vertices (0,0), (1,2), (5,4), and (6,6). R Use the transformation x = u +5v, y = 2u +4v
Find (4x + 2y)dA where R is the parallelogram with vertices (0,0), (1,2), (5,4), and (6,6). R Use the transformation x = u +5v, y = 2u +4v
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
Find the double integral:
\[ \iint_R (4x + 2y) \, dA \]
where \( R \) is the parallelogram with vertices at \( (0,0) \), \( (1,2) \), \( (5,4) \), and \( (6,6) \).
Use the transformation:
\[ x = u + 5v, \quad y = 2u + 4v \]
---
#### Explanation:
This problem requires evaluating a double integral over a given region \( R \), which in this case is a parallelogram.
The vertices of the parallelogram are:
- \( (0,0) \)
- \( (1,2) \)
- \( (5,4) \)
- \( (6,6) \)
A linear transformation is provided to simplify the region over which the integral is taken:
\[ x = u + 5v \]
\[ y = 2u + 4v \]
Using this transformation, the problem asks to compute the integral in the new coordinates. The function to be integrated is \( 4x + 2y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2bc9f06-7337-4a04-9aea-9ce8610fc600%2Faacb61f7-2c3c-4170-bbd1-8cc4ec994698%2Fx339ele_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the double integral:
\[ \iint_R (4x + 2y) \, dA \]
where \( R \) is the parallelogram with vertices at \( (0,0) \), \( (1,2) \), \( (5,4) \), and \( (6,6) \).
Use the transformation:
\[ x = u + 5v, \quad y = 2u + 4v \]
---
#### Explanation:
This problem requires evaluating a double integral over a given region \( R \), which in this case is a parallelogram.
The vertices of the parallelogram are:
- \( (0,0) \)
- \( (1,2) \)
- \( (5,4) \)
- \( (6,6) \)
A linear transformation is provided to simplify the region over which the integral is taken:
\[ x = u + 5v \]
\[ y = 2u + 4v \]
Using this transformation, the problem asks to compute the integral in the new coordinates. The function to be integrated is \( 4x + 2y \).
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