Evaluate: $(6xy + y2) dx + 3x²dy. C is the boundary of the graphs of y = 8x3 from (1,8) to (0, 0) followed by a line segments from (0, 0) to (1,8).
Evaluate: $(6xy + y2) dx + 3x²dy. C is the boundary of the graphs of y = 8x3 from (1,8) to (0, 0) followed by a line segments from (0, 0) to (1,8).
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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I need help with #4
![### Text Transcription for Educational Website:
**Problem Statement:**
4. Evaluate the integral:
\[
\oint (6xy + y^2) \, dx + 3x^2 \, dy
\]
C is the boundary of the graphs of \( y = 8x^3 \) from (1, 8) to (0, 0) followed by a line segment from (0, 0) to (1, 8).
**Graph/Diagram Explanation:**
The problem describes a curve C composed of two parts:
1. The graph of the function \( y = 8x^3 \), which extends from the point (1, 8) to the origin (0, 0).
2. A straight line segment that connects the origin (0, 0) to the point (1, 8).
The integral is a line integral evaluated over the closed curve C, combining the two defined paths. It involves terms with differentials \( dx \) and \( dy \), which suggests it is related to a vector field or a Green's Theorem application in a plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca92bf5f-8e35-4ea7-bd85-d600dcb720b2%2F7166de0e-1e32-4841-8455-652c789bb387%2Fe7urvp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Text Transcription for Educational Website:
**Problem Statement:**
4. Evaluate the integral:
\[
\oint (6xy + y^2) \, dx + 3x^2 \, dy
\]
C is the boundary of the graphs of \( y = 8x^3 \) from (1, 8) to (0, 0) followed by a line segment from (0, 0) to (1, 8).
**Graph/Diagram Explanation:**
The problem describes a curve C composed of two parts:
1. The graph of the function \( y = 8x^3 \), which extends from the point (1, 8) to the origin (0, 0).
2. A straight line segment that connects the origin (0, 0) to the point (1, 8).
The integral is a line integral evaluated over the closed curve C, combining the two defined paths. It involves terms with differentials \( dx \) and \( dy \), which suggests it is related to a vector field or a Green's Theorem application in a plane.
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