We want to evaluate P, F · dr, where F(x, y) = and C is the circle x² +y² = 1, oriented clockwise. Which of the following integrals results from applying Green's theorem to this problem? L L rdr de OL Lx² + y°dx dy O- KL r²dr d0 2n o" 6 r?dr d0
We want to evaluate P, F · dr, where F(x, y) = and C is the circle x² +y² = 1, oriented clockwise. Which of the following integrals results from applying Green's theorem to this problem? L L rdr de OL Lx² + y°dx dy O- KL r²dr d0 2n o" 6 r?dr d0
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...
Related questions
Question
![We want to evaluate \( \oint_C \mathbf{F} \cdot d\mathbf{r} \), where
\[ \mathbf{F}(x, y) = \langle e^{2x} + x^2 y, e^{2y} - xy^2 \rangle \]
and \( C \) is the circle \( x^2 + y^2 = 1 \), oriented clockwise. Which of the following integrals results from applying Green's theorem to this problem?
1. \( \int_0^{2\pi} \int_0^1 r^3 \, dr \, d\theta \)
2. \( \int_0^1 \int_0^1 x^2 + y^2 \, dx \, dy \)
3. \( -\int_0^{\pi} \int_0^1 r^2 \, dr \, d\theta \)
4. \( \int_0^{2\pi} \int_0^1 r^2 \, dr \, d\theta \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef181dc2-fef5-43c0-ba63-c5f0d82b0b96%2Ff48f1128-e1bc-4b08-92b7-e8149d06eae1%2Fuzu7guq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We want to evaluate \( \oint_C \mathbf{F} \cdot d\mathbf{r} \), where
\[ \mathbf{F}(x, y) = \langle e^{2x} + x^2 y, e^{2y} - xy^2 \rangle \]
and \( C \) is the circle \( x^2 + y^2 = 1 \), oriented clockwise. Which of the following integrals results from applying Green's theorem to this problem?
1. \( \int_0^{2\pi} \int_0^1 r^3 \, dr \, d\theta \)
2. \( \int_0^1 \int_0^1 x^2 + y^2 \, dx \, dy \)
3. \( -\int_0^{\pi} \int_0^1 r^2 \, dr \, d\theta \)
4. \( \int_0^{2\pi} \int_0^1 r^2 \, dr \, d\theta \)
![**Evaluating Line Integrals Over Curves**
We want to evaluate the line integral \(\oint_C \mathbf{F} \cdot d\mathbf{r}\), where:
\[
\mathbf{F}(x, y) = \langle e^{2x} + x^2y, e^{2y} - xy^2 \rangle
\]
and \(C\) is the circle defined by the equation \(x^2 + y^2 = 1\), oriented clockwise. The question is: how is this integral related to the integral over the curve oriented counter-clockwise?
**Options:**
- ☐ The two integrals are the same.
- ☐ The clockwise integral is not defined; we can only integrate in a counter-clockwise direction.
- ☑ The integral over the clockwise curve is the opposite sign of the integral over the counter-clockwise curve.
- ☐ The two integrals give completely different values.
**Explanation:**
For line integrals over closed curves, reversing the direction of the curve changes the sign of the integral. Therefore, the integral over the clockwise curve will be the negative of the integral over the counter-clockwise curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef181dc2-fef5-43c0-ba63-c5f0d82b0b96%2Ff48f1128-e1bc-4b08-92b7-e8149d06eae1%2F5r6ykn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluating Line Integrals Over Curves**
We want to evaluate the line integral \(\oint_C \mathbf{F} \cdot d\mathbf{r}\), where:
\[
\mathbf{F}(x, y) = \langle e^{2x} + x^2y, e^{2y} - xy^2 \rangle
\]
and \(C\) is the circle defined by the equation \(x^2 + y^2 = 1\), oriented clockwise. The question is: how is this integral related to the integral over the curve oriented counter-clockwise?
**Options:**
- ☐ The two integrals are the same.
- ☐ The clockwise integral is not defined; we can only integrate in a counter-clockwise direction.
- ☑ The integral over the clockwise curve is the opposite sign of the integral over the counter-clockwise curve.
- ☐ The two integrals give completely different values.
**Explanation:**
For line integrals over closed curves, reversing the direction of the curve changes the sign of the integral. Therefore, the integral over the clockwise curve will be the negative of the integral over the counter-clockwise curve.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning