**Problem Statement:**Use Green's Theorem to evaluate the line integral along the given positively oriented curve.\[ \oint_{C} xe^{-5x} \, dx + \left( x^4 + 2x^2y^2 \right) \, dy \]**Note:**- $ C $ is the boundary of the region between the circles $ x^2 + y^2 = 9 $ and $ x^2 + y^2 = 25 $.### Explanation:**Green's Theorem** connects a line integral around a simple closed curve $ C $ to a double integral over the plane region $ D $ bounded by $ C $:\[\oint_{C} \left( M \, dx + N \, dy \right) = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA\]Here:- $ M(x, y) = xe^{-5x} $- $ N(x, y) = x^4 + 2x^2y^2 $**Region $ C $**: The region is an annular area (the area between two circles) with radii determined by:- The inner circle: $ x^2 + y^2 = 9 $, with radius 3.- The outer circle: $ x^2 + y^2 = 25 $, with radius 5.**Graphs/Diagrams:**- Assume two concentric circles are drawn on a coordinate plane.- The inner circle has a radius of 3 and the outer circle has a radius of 5.- The region $ C $ is the shaded area between these two circles.To solve, apply Green's Theorem to convert the line integral into a double integral over the annular region. Calculate $ \frac{\partial N}{\partial x} $ and $ \frac{\partial M}{\partial y} $, and integrate over the given region to find the area.

Answered: Image /qna-images/answer/ddde3b41-2ffa-4b95-9c45-15df7aabd850.jpg

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. -5x хе (x* + 2x3y²) dy dx + Cis the boundary of the region between the circles x2 + y? = 9 and x2 + y2 = 25 %3D

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. -5x хе (x* + 2x3y²) dy dx + Cis the boundary of the region between the circles x2 + y? = 9 and x2 + y2 = 25 %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A: Answer

Question

**Problem Statement:**

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

\[
\oint_{C} xe^{-5x} \, dx + \left( x^4 + 2x^2y^2 \right) \, dy
\]

**Note:**
- $ C $ is the boundary of the region between the circles $ x^2 + y^2 = 9 $ and $ x^2 + y^2 = 25 $.

### Explanation:

**Green's Theorem** connects a line integral around a simple closed curve $ C $ to a double integral over the plane region $ D $ bounded by $ C $:

\[
\oint_{C} \left( M \, dx + N \, dy \right) = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA
\]

Here:

- $ M(x, y) = xe^{-5x} $
- $ N(x, y) = x^4 + 2x^2y^2 $

**Region $ C $**: The region is an annular area (the area between two circles) with radii determined by:

- The inner circle: $ x^2 + y^2 = 9 $, with radius 3.
- The outer circle: $ x^2 + y^2 = 25 $, with radius 5.

**Graphs/Diagrams:**

- Assume two concentric circles are drawn on a coordinate plane.
- The inner circle has a radius of 3 and the outer circle has a radius of 5.
- The region $ C $ is the shaded area between these two circles.

To solve, apply Green's Theorem to convert the line integral into a double integral over the annular region. Calculate $ \frac{\partial N}{\partial x} $ and $ \frac{\partial M}{\partial y} $, and integrate over the given region to find the area.

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