Use Stoke's Theorem to evaluate F. dr where F = (3x + 7y, 2y + 3z, 10z + 6x) and C is the triangle with vertices (2, 0, 0), (0, 6, 0) and (0, 0, 12) orientated so that the vertices are traversed in the specified order. The line integral equals Submit Answer Tries 0/30
Use Stoke's Theorem to evaluate F. dr where F = (3x + 7y, 2y + 3z, 10z + 6x) and C is the triangle with vertices (2, 0, 0), (0, 6, 0) and (0, 0, 12) orientated so that the vertices are traversed in the specified order. The line integral equals Submit Answer Tries 0/30
Calculus: Early Transcendentals
8th Edition
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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![### Applying Stokes' Theorem to Evaluate a Line Integral
Use Stokes' Theorem to evaluate the line integral
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} \]
where \( \mathbf{F} = \langle 3x + 7y, 2y + 3z, 10z + 6x \rangle \) and C is the triangle with vertices (2, 0, 0), (0, 6, 0), and (0, 0, 12) oriented so that the vertices are traversed in the specified order.
### Problem Details
- **Vector Field**: \( \mathbf{F} = \langle 3x + 7y, 2y + 3z, 10z + 6x \rangle \)
- **Curve C**: Defined by the triangle with vertices:
1. \( (2, 0, 0) \)
2. \( (0, 6, 0) \)
3. \( (0, 0, 12) \)
- **Orientation**: The vertices are traversed in the order as given.
### Instructions
To solve this problem using Stokes' Theorem:
1. **Stokes' Theorem Statement**: Stokes' Theorem relates a surface integral over a surface \( S \) to a line integral over the boundary curve \( C \):
\[
\int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
\]
2. **Calculate the Curl**: Find \( \nabla \times \mathbf{F} \).
3. **Surface Integral**: Parameterize the surface \( S \) and express the surface integral.
4. **Compute the Integral**: Evaluate the surface integral to find the value of the line integral.
### Input Your Solution
The line integral equals __________
(*Note: If this problem is part of a homework or exam, ensure you perform the detailed calculations as required.*)
### Interactive Section
Submit your calculated answer in the box below and check the correctness of your solution:
\[ \text{The line integral equals } \boxed{} \]
Button: **Submit](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2Fbd287efe-b987-46e8-889c-d3371d48cb65%2F72bj2lg_processed.png&w=3840&q=75)
Transcribed Image Text:### Applying Stokes' Theorem to Evaluate a Line Integral
Use Stokes' Theorem to evaluate the line integral
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} \]
where \( \mathbf{F} = \langle 3x + 7y, 2y + 3z, 10z + 6x \rangle \) and C is the triangle with vertices (2, 0, 0), (0, 6, 0), and (0, 0, 12) oriented so that the vertices are traversed in the specified order.
### Problem Details
- **Vector Field**: \( \mathbf{F} = \langle 3x + 7y, 2y + 3z, 10z + 6x \rangle \)
- **Curve C**: Defined by the triangle with vertices:
1. \( (2, 0, 0) \)
2. \( (0, 6, 0) \)
3. \( (0, 0, 12) \)
- **Orientation**: The vertices are traversed in the order as given.
### Instructions
To solve this problem using Stokes' Theorem:
1. **Stokes' Theorem Statement**: Stokes' Theorem relates a surface integral over a surface \( S \) to a line integral over the boundary curve \( C \):
\[
\int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
\]
2. **Calculate the Curl**: Find \( \nabla \times \mathbf{F} \).
3. **Surface Integral**: Parameterize the surface \( S \) and express the surface integral.
4. **Compute the Integral**: Evaluate the surface integral to find the value of the line integral.
### Input Your Solution
The line integral equals __________
(*Note: If this problem is part of a homework or exam, ensure you perform the detailed calculations as required.*)
### Interactive Section
Submit your calculated answer in the box below and check the correctness of your solution:
\[ \text{The line integral equals } \boxed{} \]
Button: **Submit
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