Use the indicated change of variables to evaluate the double integral. √√√√x²x= y(x - y) dA X=U+V y = u
Use the indicated change of variables to evaluate the double integral. √√√√x²x= y(x - y) dA X=U+V y = u
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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![### Topic: Evaluating Double Integrals Using Change of Variables
#### Problem Statement:
Use the indicated change of variables to evaluate the double integral.
\[
\int_R \int y(x-y) \, dA
\]
#### Change of Variables:
- \( x = u + v \)
- \( y = u \)
### Explanation of the Graph:
The graph shows a parallelogram on the coordinate plane, which represents the region \( R \) in the \( xy \)-plane over which the double integral is evaluated.
#### Key Points:
- The vertices of the parallelogram are labeled as follows:
- Bottom-left at \( (0, 0) \)
- Bottom-right at \( (8, 0) \)
- Top-left at \( (7, 7) \)
- Top-right at \( (15, 7) \)
#### Axes:
- The horizontal axis is labeled as \( x \) and ranges from \(-2\) to \(15\).
- The vertical axis is labeled as \( y \) and ranges from \( -2 \) to \(10\).
### Interpretation:
The given transformation \( x = u + v \) and \( y = u \) needs to be applied to assess how this corresponds to the given region \( R \) and to facilitate the calculation of the integral. The geometry of the region, shown as a parallelogram, is a key aspect in setting up limits for integration after variables are changed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa48f7eb-bce8-4ea0-bf8b-7482c9c56a7c%2Fe2f28e3c-a544-4aee-b06f-b6202b7049dd%2Ftktebf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Evaluating Double Integrals Using Change of Variables
#### Problem Statement:
Use the indicated change of variables to evaluate the double integral.
\[
\int_R \int y(x-y) \, dA
\]
#### Change of Variables:
- \( x = u + v \)
- \( y = u \)
### Explanation of the Graph:
The graph shows a parallelogram on the coordinate plane, which represents the region \( R \) in the \( xy \)-plane over which the double integral is evaluated.
#### Key Points:
- The vertices of the parallelogram are labeled as follows:
- Bottom-left at \( (0, 0) \)
- Bottom-right at \( (8, 0) \)
- Top-left at \( (7, 7) \)
- Top-right at \( (15, 7) \)
#### Axes:
- The horizontal axis is labeled as \( x \) and ranges from \(-2\) to \(15\).
- The vertical axis is labeled as \( y \) and ranges from \( -2 \) to \(10\).
### Interpretation:
The given transformation \( x = u + v \) and \( y = u \) needs to be applied to assess how this corresponds to the given region \( R \) and to facilitate the calculation of the integral. The geometry of the region, shown as a parallelogram, is a key aspect in setting up limits for integration after variables are changed.
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