Use Green's Theorem to evaluate the line integral of F=<3y+1, 4x²+3> over C which is the boundary of the rectangular region with vertices (0,0),(4,0),(4,2) and (0,2), oriented counterclockwise.
Use Green's Theorem to evaluate the line integral of F=<3y+1, 4x²+3> over C which is the boundary of the rectangular region with vertices (0,0),(4,0),(4,2) and (0,2), oriented counterclockwise.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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![**Green’s Theorem Application**
*Topic: Evaluating Line Integrals.*
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**Problem Statement:**
Use Green’s Theorem to evaluate the line integral of **F = <3y + 1, 4x^2 + 3>** over **C**, which is the boundary of the rectangular region with vertices **(0,0), (4,0), (4,2),** and **(0,2)**, oriented counterclockwise.
---
**Explanation:**
Green’s Theorem provides a relationship between a line integral around a simple closed curve **C** and a double integral over the plane region **D** bounded by **C**. It is often used in the context of evaluating a line integral more easily.
According to Green’s Theorem:
\[ \oint_{C} (P dx + Q dy) = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]
In this problem, the vector field **F** is given as **F = <3y + 1, 4x^2 + 3>**, meaning:
- **P(x,y) = 3y + 1**
- **Q(x,y) = 4x^2 + 3**
The boundary **C** is defined as the rectangular region with vertices **(0,0)**, **(4,0)**, **(4,2)**, and **(0,2)**.
1. **Compute the Partial Derivatives:**
- \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (4x^2 + 3) = 8x \)
- \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (3y + 1) = 3 \)
2. **Substitute into Green’s Theorem:**
\[ \iint_{D} (8x - 3) dA \]
3. **Set up the Double Integral over Region D:**
The region **D** is a rectangle defined by \(0 \leq x \leq 4\) and \(0 \leq y \leq 2\):
\[ \int_{0}^{2} \int_{0}^{4} (8x -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bb816e9-0766-4c54-994c-f75fc6d9bab1%2F0dbda278-d178-4dec-87e6-60816bfc84d3%2Fbg0vvf8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Green’s Theorem Application**
*Topic: Evaluating Line Integrals.*
---
**Problem Statement:**
Use Green’s Theorem to evaluate the line integral of **F = <3y + 1, 4x^2 + 3>** over **C**, which is the boundary of the rectangular region with vertices **(0,0), (4,0), (4,2),** and **(0,2)**, oriented counterclockwise.
---
**Explanation:**
Green’s Theorem provides a relationship between a line integral around a simple closed curve **C** and a double integral over the plane region **D** bounded by **C**. It is often used in the context of evaluating a line integral more easily.
According to Green’s Theorem:
\[ \oint_{C} (P dx + Q dy) = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]
In this problem, the vector field **F** is given as **F = <3y + 1, 4x^2 + 3>**, meaning:
- **P(x,y) = 3y + 1**
- **Q(x,y) = 4x^2 + 3**
The boundary **C** is defined as the rectangular region with vertices **(0,0)**, **(4,0)**, **(4,2)**, and **(0,2)**.
1. **Compute the Partial Derivatives:**
- \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (4x^2 + 3) = 8x \)
- \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (3y + 1) = 3 \)
2. **Substitute into Green’s Theorem:**
\[ \iint_{D} (8x - 3) dA \]
3. **Set up the Double Integral over Region D:**
The region **D** is a rectangle defined by \(0 \leq x \leq 4\) and \(0 \leq y \leq 2\):
\[ \int_{0}^{2} \int_{0}^{4} (8x -
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