Use Green's Theorem to evaluate F. dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y cos(x) - xy sin(x), xy + x cos(x)), C is the triangle from (0, 0) to (0, 12) to (3, 0) to (0, 0)
Use Green's Theorem to evaluate F. dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y cos(x) - xy sin(x), xy + x cos(x)), C is the triangle from (0, 0) to (0, 12) to (3, 0) to (0, 0)
Calculus: Early Transcendentals
8th Edition
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Use Green's Theorem to Evaluate the Line Integral**
Evaluate the line integral using Green's Theorem:
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} \]
*(Check the orientation of the curve before applying the theorem.)*
**Given:**
\[ \mathbf{F}(x, y) = \left( y \cos(x) - xy \sin(x), \, xy + x \cos(x) \right) \]
**Curve \( C \):**
\( C \) is the triangle with vertices at \( (0, 0) \), \( (0, 12) \), and \( (3, 0) \).
---
In this problem, we are asked to apply Green's Theorem to evaluate the line integral over a triangular path. Green's Theorem relates the line integral around a simple closed curve \( C \) to a double integral over the region \( D \) bounded by \( C \). The theorem is expressed as:
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \]
where \( \mathbf{F}(x, y) = (M(x, y), N(x, y)) \).
**Step-by-step Overview:**
1. **Identify Curve Orientation:**
- Ensure that the curve \( C \) is oriented counterclockwise.
2. **Apply Green's Theorem:**
- Determine \( M \) and \( N \) from the vector field \( \mathbf{F}(x, y) = (M(x, y), N(x, y)) \).
- Compute \( \frac{\partial N}{\partial x} \) and \( \frac{\partial M}{\partial y} \).
- Evaluate the double integral over the region \( D \) bounded by \( C \).
This structured approach will guide in computing the line integral by transforming it into a simpler double integral using Green's Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83a2909f-e771-46af-ac38-b46aeee99cc8%2F3ddfa639-2572-4fa1-9b7b-3f9e33bc5528%2F27y8l3t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Use Green's Theorem to Evaluate the Line Integral**
Evaluate the line integral using Green's Theorem:
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} \]
*(Check the orientation of the curve before applying the theorem.)*
**Given:**
\[ \mathbf{F}(x, y) = \left( y \cos(x) - xy \sin(x), \, xy + x \cos(x) \right) \]
**Curve \( C \):**
\( C \) is the triangle with vertices at \( (0, 0) \), \( (0, 12) \), and \( (3, 0) \).
---
In this problem, we are asked to apply Green's Theorem to evaluate the line integral over a triangular path. Green's Theorem relates the line integral around a simple closed curve \( C \) to a double integral over the region \( D \) bounded by \( C \). The theorem is expressed as:
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \]
where \( \mathbf{F}(x, y) = (M(x, y), N(x, y)) \).
**Step-by-step Overview:**
1. **Identify Curve Orientation:**
- Ensure that the curve \( C \) is oriented counterclockwise.
2. **Apply Green's Theorem:**
- Determine \( M \) and \( N \) from the vector field \( \mathbf{F}(x, y) = (M(x, y), N(x, y)) \).
- Compute \( \frac{\partial N}{\partial x} \) and \( \frac{\partial M}{\partial y} \).
- Evaluate the double integral over the region \( D \) bounded by \( C \).
This structured approach will guide in computing the line integral by transforming it into a simpler double integral using Green's Theorem.
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