Use Green's Theorem to evaluate the line integral [(y-x) dx + (2x−y) dy for the given path. C: boundary of the region lying inside the semicircle y = 49 and outside the semicircle y - 25
Use Green's Theorem to evaluate the line integral [(y-x) dx + (2x−y) dy for the given path. C: boundary of the region lying inside the semicircle y = 49 and outside the semicircle y - 25
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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![### Using Green's Theorem for Line Integrals
#### Task:
Apply Green’s Theorem to evaluate the line integral:
\[
\oint_C (y - x) \, dx + (2x - y) \, dy
\]
for the specified path \( C \).
#### Description of Path \( C \):
- The path \( C \) is described as the boundary of the region lying:
- Inside the semicircle \( y = \sqrt{49 - x^2} \)
- Outside the semicircle \( y = \sqrt{25 - x^2} \)
This involves evaluating the line integral over a closed path \( C \) that is bounded by these semicircles, using Green’s Theorem to transform the line integral into a double integral over the region enclosed by \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53d7c283-d7c7-48d5-8c8d-bb71f4fe00d9%2Fe2c7bcaf-4acc-4d20-bc25-a6b51a1ae98d%2Fizbzwow_processed.png&w=3840&q=75)
Transcribed Image Text:### Using Green's Theorem for Line Integrals
#### Task:
Apply Green’s Theorem to evaluate the line integral:
\[
\oint_C (y - x) \, dx + (2x - y) \, dy
\]
for the specified path \( C \).
#### Description of Path \( C \):
- The path \( C \) is described as the boundary of the region lying:
- Inside the semicircle \( y = \sqrt{49 - x^2} \)
- Outside the semicircle \( y = \sqrt{25 - x^2} \)
This involves evaluating the line integral over a closed path \( C \) that is bounded by these semicircles, using Green’s Theorem to transform the line integral into a double integral over the region enclosed by \( C \).
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