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Evaluate O (6y+x)dx+(y+2x)dy where C is the circle (x-2)²+(y-3)² = 4 Without using Green's Theorem.

Evaluate O (6y+x)dx+(y+2x)dy where C is the circle (x-2)²+(y-3)² = 4 Without using Green's Theorem.

Advanced Engineering Mathematics
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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Question
**Problem 2:** Evaluate \(\oint_C (6y + x) \, dx + (y + 2x) \, dy\) where \(C\) is the circle \((x - 2)^2 + (y - 3)^2 = 4\). Without using Green's Theorem. --- **Explanation:** - **Integral:** The problem involves evaluating a line integral \(\oint_C\) of the vector field \((6y + x, y + 2x)\) along the curve \(C\). - **Curve \(C\):** The curve is described by the equation \((x - 2)^2 + (y - 3)^2 = 4\), which represents a circle centered at \((2, 3)\) with a radius of 2. - **Method Restriction:** The evaluation must be completed without using Green’s Theorem, which usually relates a line integral around a simple closed curve \(C\) and a double integral over the plane region \(D\) bounded by \(C\).
Transcribed Image Text:**Problem 2:** Evaluate \(\oint_C (6y + x) \, dx + (y + 2x) \, dy\) where \(C\) is the circle \((x - 2)^2 + (y - 3)^2 = 4\). Without using Green's Theorem. --- **Explanation:** - **Integral:** The problem involves evaluating a line integral \(\oint_C\) of the vector field \((6y + x, y + 2x)\) along the curve \(C\). - **Curve \(C\):** The curve is described by the equation \((x - 2)^2 + (y - 3)^2 = 4\), which represents a circle centered at \((2, 3)\) with a radius of 2. - **Method Restriction:** The evaluation must be completed without using Green’s Theorem, which usually relates a line integral around a simple closed curve \(C\) and a double integral over the plane region \(D\) bounded by \(C\).
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