Use Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. $c (5x + sinh y)dy − (3y² + arctan x²) dx, where C is the boundary of the square with vertices (1, 3), (4, 3), (4, 6), and (1, 6). $c с (Type an exact answer.) - (3y² + arctan x² (5x + sinh y)dy – nx²) dx dx = (
Use Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. $c (5x + sinh y)dy − (3y² + arctan x²) dx, where C is the boundary of the square with vertices (1, 3), (4, 3), (4, 6), and (1, 6). $c с (Type an exact answer.) - (3y² + arctan x² (5x + sinh y)dy – nx²) dx dx = (
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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![# Line Integral Evaluation Using Green's Theorem
**Problem Statement:**
Utilize Green’s Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise.
\[
\oint_C \left(5x + \sinh y\right) dy - \left(3y^2 + \arctan x^2 \right) dx,
\]
where \( C \) is the boundary of the square with vertices \((1, 3)\), \((4, 3)\), \((4, 6)\), and \((1, 6)\).
**Solution:**
\[
\oint_C \left(5x + \sinh y\right) dy - \left(3y^2 + \arctan x^2 \right) dx = \square
\]
*(Type an exact answer.)*
---
**Explanation:**
This task involves evaluating a line integral using Green's Theorem, which relates a line integral around a simple closed curve \( C \) to a double integral over the plane region \( D \) bounded by \( C \). The theorem is applicable here because the region is a square defined by the given vertices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2c370c8-f220-47ec-b561-9f6a07b2c79a%2F3e5dec0d-2f13-4c73-adbf-7180c615b5b7%2Fc5n73j_processed.png&w=3840&q=75)
Transcribed Image Text:# Line Integral Evaluation Using Green's Theorem
**Problem Statement:**
Utilize Green’s Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise.
\[
\oint_C \left(5x + \sinh y\right) dy - \left(3y^2 + \arctan x^2 \right) dx,
\]
where \( C \) is the boundary of the square with vertices \((1, 3)\), \((4, 3)\), \((4, 6)\), and \((1, 6)\).
**Solution:**
\[
\oint_C \left(5x + \sinh y\right) dy - \left(3y^2 + \arctan x^2 \right) dx = \square
\]
*(Type an exact answer.)*
---
**Explanation:**
This task involves evaluating a line integral using Green's Theorem, which relates a line integral around a simple closed curve \( C \) to a double integral over the plane region \( D \) bounded by \( C \). The theorem is applicable here because the region is a square defined by the given vertices.
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