Consider the following line integral. fxy dx + x²y³ xy dx + x²y³ dy, C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 4)
Consider the following line integral. fxy dx + x²y³ xy dx + x²y³ dy, C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
A) Evaluate the given line
![### Line Integral Problem
Consider the following line integral:
\[
\oint_C xy \, dx + x^2 y^3 \, dy
\]
where \( C \) is the counterclockwise path around the triangle with vertices at \((0, 0)\), \((1, 0)\), and \((1, 4)\).
#### Explanation:
- **Line Integral:** This expression is a line integral over the curve \( C \). The operation involves integrating along a path in the xy-plane.
- **Components:**
- \( xy \, dx \): This term integrates the product of \( x \) and \( y \) with respect to \( x \).
- \( x^2 y^3 \, dy \): This term integrates the product of \( x^2 \) and \( y^3 \) with respect to \( y \).
- **Path \( C \):**
- The path is a triangle with vertices at the specified coordinates. The integration is performed counterclockwise along this triangular path.
The problem involves calculating the total value of the integral along this defined triangular path.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d1a45f8-3ba2-4624-865f-f36d23213ebc%2Fb82b7fa7-dd9b-4b99-835f-d2686f7ef4d5%2Fqzllhp_processed.png&w=3840&q=75)
Transcribed Image Text:### Line Integral Problem
Consider the following line integral:
\[
\oint_C xy \, dx + x^2 y^3 \, dy
\]
where \( C \) is the counterclockwise path around the triangle with vertices at \((0, 0)\), \((1, 0)\), and \((1, 4)\).
#### Explanation:
- **Line Integral:** This expression is a line integral over the curve \( C \). The operation involves integrating along a path in the xy-plane.
- **Components:**
- \( xy \, dx \): This term integrates the product of \( x \) and \( y \) with respect to \( x \).
- \( x^2 y^3 \, dy \): This term integrates the product of \( x^2 \) and \( y^3 \) with respect to \( y \).
- **Path \( C \):**
- The path is a triangle with vertices at the specified coordinates. The integration is performed counterclockwise along this triangular path.
The problem involves calculating the total value of the integral along this defined triangular path.
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