to evaluate the line 12) Use Gren's Theorem integead along the posticy orientrch curre Jxy²dx +2x²ydy Cis the triangle with verticea (0,0), (2,2),62,4) 4. 2.
to evaluate the line 12) Use Gren's Theorem integead along the posticy orientrch curre Jxy²dx +2x²ydy Cis the triangle with verticea (0,0), (2,2),62,4) 4. 2.
Calculus: Early Transcendentals
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ISBN:9781285741550
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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![**Using Green's Theorem to Evaluate a Line Integral:**
**Problem Statement:**
Use Green's Theorem to evaluate the line integral along the positively oriented curve
\[ \oint_C x^2 \, dx + 2x^2 y \, dy \]
where \( C \) is the triangle with vertices \((0,0)\), \((2,2)\), and \((2,4)\).
**Graph Description:**
The provided graph depicts a triangle with the following vertices:
- \((0,0)\)
- \((2,2)\)
- \((2,4)\)
The triangle is shown on the Cartesian coordinate system, with one vertex at the origin and the other two vertices lying along the line formed by \( x = 2 \) at points (2, 2) and (2, 4). The triangle is oriented in a counterclockwise direction, which indicates that the curve \( C \) is positively oriented.
The x-axis intersects the triangle at points \((0,0)\) and \((2,2)\), while the y-axis extends up to 4 to accommodate the highest point of the triangle at \((2,4)\).
**Steps to be followed to solve using Green's Theorem:**
1. Identify the function components for \( P(x,y) \) and \( Q(x,y) \). In this case:
- \( P(x,y) = x^2 \)
- \( Q(x,y) = 2x^2 y \)
2. Compute the partial derivatives needed for Green's Theorem:
- \[ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (2x^2 y) = 4xy \]
- \[ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (x^2) = 0 \]
3. Plug these into Green's Theorem integral:
\[ \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]
- Therefore:
\[ \oint_C x^2 \, dx + 2x^2 y \, dy = \iint_D (4xy -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe613d3f6-ed3c-4abe-81e0-270d8219b401%2Fd73e5942-c58c-49f9-aae4-dbdf64b7e55d%2Fxl7mmy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Green's Theorem to Evaluate a Line Integral:**
**Problem Statement:**
Use Green's Theorem to evaluate the line integral along the positively oriented curve
\[ \oint_C x^2 \, dx + 2x^2 y \, dy \]
where \( C \) is the triangle with vertices \((0,0)\), \((2,2)\), and \((2,4)\).
**Graph Description:**
The provided graph depicts a triangle with the following vertices:
- \((0,0)\)
- \((2,2)\)
- \((2,4)\)
The triangle is shown on the Cartesian coordinate system, with one vertex at the origin and the other two vertices lying along the line formed by \( x = 2 \) at points (2, 2) and (2, 4). The triangle is oriented in a counterclockwise direction, which indicates that the curve \( C \) is positively oriented.
The x-axis intersects the triangle at points \((0,0)\) and \((2,2)\), while the y-axis extends up to 4 to accommodate the highest point of the triangle at \((2,4)\).
**Steps to be followed to solve using Green's Theorem:**
1. Identify the function components for \( P(x,y) \) and \( Q(x,y) \). In this case:
- \( P(x,y) = x^2 \)
- \( Q(x,y) = 2x^2 y \)
2. Compute the partial derivatives needed for Green's Theorem:
- \[ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (2x^2 y) = 4xy \]
- \[ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (x^2) = 0 \]
3. Plug these into Green's Theorem integral:
\[ \oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]
- Therefore:
\[ \oint_C x^2 \, dx + 2x^2 y \, dy = \iint_D (4xy -
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