Find (3x + 2y)dA where R is the parallelogram with vertices (0,0), (4,5), (-3,1), and (1,6). Use the transformation = 4u3v, y = 5u + v.
Find (3x + 2y)dA where R is the parallelogram with vertices (0,0), (4,5), (-3,1), and (1,6). Use the transformation = 4u3v, y = 5u + v.
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 \(\iint\limits_{R} (3x + 2y) \, dA\) where \(R\) is the parallelogram with vertices \((0,0)\), \((4,5)\), \((-3, -1)\), and \((1,6)\).
**Transformation Technique:**
Use the transformation:
\[ x = 4u - 3v, \quad y = 5u + v \]
**Instructions for Students:**
- Understand the given transformation as a method to simplify the integration process over a parallelogram.
- Rewrite the given integration problem in terms of the new variables \(u\) and \(v\) using the transformation equations.
- Calculate the Jacobian of the transformation and adjust the integral accordingly.
- Determine the new limits of integration based on the transformed region.
This exercise aims to illustrate the application of linear transformations in multiple integrals to simplify the domain of integration, highlighting the computation involving Jacobians.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2Fe1ad8369-29af-4c3f-a7dd-74487da3cc27%2Fs4cndi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the double integral \(\iint\limits_{R} (3x + 2y) \, dA\) where \(R\) is the parallelogram with vertices \((0,0)\), \((4,5)\), \((-3, -1)\), and \((1,6)\).
**Transformation Technique:**
Use the transformation:
\[ x = 4u - 3v, \quad y = 5u + v \]
**Instructions for Students:**
- Understand the given transformation as a method to simplify the integration process over a parallelogram.
- Rewrite the given integration problem in terms of the new variables \(u\) and \(v\) using the transformation equations.
- Calculate the Jacobian of the transformation and adjust the integral accordingly.
- Determine the new limits of integration based on the transformed region.
This exercise aims to illustrate the application of linear transformations in multiple integrals to simplify the domain of integration, highlighting the computation involving Jacobians.
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