Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**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.](https://content.bartleby.com/qna-images/question/73312b24-b9c5-4faa-988d-248cbc876a5a/ddde3b41-2ffa-4b95-9c45-15df7aabd850/pkjy9q_processed.png)
Transcribed Image Text:**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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