Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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.](https://content.bartleby.com/qna-images/question/83a2909f-e771-46af-ac38-b46aeee99cc8/3ddfa639-2572-4fa1-9b7b-3f9e33bc5528/27y8l3t_processed.jpeg)
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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