Use the given transformation to evaluate the integral. (12x + 12y) dA R, where R is the parallelogram with vertices (−2, 4), (2, −4), (4, −2), and (0, 6); x = 13(u + v), y = 13(v − 2u) **Transcription for Educational Website:**---Use the given transformation to evaluate the integral.\[\iint\limits_{R} (12x + 12y) \, dA,\]where \( R \) is the parallelogram with vertices \((-2, 4)\), \( (2, -4)\), \( (4, -2)\), and \( (0, 6)\); the transformation is given by:\[ x = \frac{1}{3} (u + v), \quad y = \frac{1}{3} (v - 2u) \]**Explanation of the Transformation:**This problem involves evaluating a double integral over a region \( R \) defined as a parallelogram with specific vertices. The transformation equations provided are meant to switch from \( (x, y) \) coordinates to new \( (u, v) \) coordinates to simplify the integration process.1. **Vertices of the Parallelogram:** - The vertices of the parallelogram are: \( (-2, 4) \), \( (2, -4) \), \( (4, -2) \), and \( (0, 6) \).2. **Transformation Equations:** - These equations transform the \( (x, y) \) coordinates to \( (u, v) \) coordinates: - \( x = \frac{1}{3} (u + v) \) - \( y = \frac{1}{3} (v - 2u) \)The use of these transformations facilitates easier computation of the integral over the region by aligning the region \( R \) to potentially simple bounds in the \( (u, v) \) plane.--- **Note:** If you have any questions about how to apply the transformation or evaluate the integral, please refer to your calculus textbook or contact your instructor for further assistance.

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Answered: Use the given transformation to…

Use the given transformation to evaluate the integral. J (12x+12y) da, where R is the parallelogram with vertices (-2, 4), (2, -4), (4, -2), and (0, 6); x = X (u + v), y = (v - 2u)

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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Question

Use the given transformation to evaluate the integral.

	(12x + 12y) dA

R

where R is the parallelogram with vertices

(−2, 4),

(2, −4),

(4, −2),

and

(0, 6); x =

(u + v), y =

(v − 2u)

**Transcription for Educational Website:**

---

Use the given transformation to evaluate the integral.

\[
\iint\limits_{R} (12x + 12y) \, dA,
\]

where \( R \) is the parallelogram with vertices \((-2, 4)\), \( (2, -4)\), \( (4, -2)\), and \( (0, 6)\);

the transformation is given by:

\[
x = \frac{1}{3} (u + v), \quad y = \frac{1}{3} (v - 2u)
\]

**Explanation of the Transformation:**

This problem involves evaluating a double integral over a region \( R \) defined as a parallelogram with specific vertices. The transformation equations provided are meant to switch from \( (x, y) \) coordinates to new \( (u, v) \) coordinates to simplify the integration process.

1. **Vertices of the Parallelogram:**
- The vertices of the parallelogram are: \( (-2, 4) \), \( (2, -4) \), \( (4, -2) \), and \( (0, 6) \).

2. **Transformation Equations:**
- These equations transform the \( (x, y) \) coordinates to \( (u, v) \) coordinates:
- \( x = \frac{1}{3} (u + v) \)
- \( y = \frac{1}{3} (v - 2u) \)

The use of these transformations facilitates easier computation of the integral over the region by aligning the region \( R \) to potentially simple bounds in the \( (u, v) \) plane.

---

**Note:** If you have any questions about how to apply the transformation or evaluate the integral, please refer to your calculus textbook or contact your instructor for further assistance.

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