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.

Answered: Image /qna-images/answer/f48f1128-e1bc-4b08-92b7-e8149d06eae1.jpg

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...

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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 $

**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.

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