Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Problem 2: Evaluate the Line Integral
Evaluate the line integral
\[
\oint_{C} (x^4y^5 - 2y)dx + (3x + x^5y^4)dy
\]
where \( C \) is the curve given below oriented in a clockwise direction.
### Explanation of the Graph:
The graph displays a piecewise linear closed curve \( C \) on the xy-plane. The curve consists of straight line segments forming a step-like path. Below is a detailed description of the curve's path:
- **Starting Point**: The curve begins at point (0,0).
- **Segment 1**: Moves horizontally from (0, 0) to (2, 0).
- **Segment 2**: Moves vertically from (2, 0) to (2, 3).
- **Segment 3**: Moves horizontally from (2, 3) to (1, 3).
- **Segment 4**: Moves vertically from (1, 3) to (1, 1).
- **Segment 5**: Moves horizontally from (1, 1) to (0, 1).
- **Segment 6**: Moves vertically from (0, 1) back to the starting point (0, 0).
The curve forms a non-standard, step-shaped path with clear vertex points at each directional change, as denoted by the grid lines on the plane. The axes are labeled with x and y, and grid lines mark the integers from -2 to 2 on the x-axis and from 0 to 4 on the y-axis.
Note: The orientation of the curve is in a clockwise direction, which is important for evaluating the line integral.](https://content.bartleby.com/qna-images/question/77548912-c51c-4c9d-8b51-f3905a3bec75/3ecfceca-0fd3-4f25-9be2-91c34b0bd88a/5us6zhq_processed.png)
Transcribed Image Text:### Problem 2: Evaluate the Line Integral
Evaluate the line integral
\[
\oint_{C} (x^4y^5 - 2y)dx + (3x + x^5y^4)dy
\]
where \( C \) is the curve given below oriented in a clockwise direction.
### Explanation of the Graph:
The graph displays a piecewise linear closed curve \( C \) on the xy-plane. The curve consists of straight line segments forming a step-like path. Below is a detailed description of the curve's path:
- **Starting Point**: The curve begins at point (0,0).
- **Segment 1**: Moves horizontally from (0, 0) to (2, 0).
- **Segment 2**: Moves vertically from (2, 0) to (2, 3).
- **Segment 3**: Moves horizontally from (2, 3) to (1, 3).
- **Segment 4**: Moves vertically from (1, 3) to (1, 1).
- **Segment 5**: Moves horizontally from (1, 1) to (0, 1).
- **Segment 6**: Moves vertically from (0, 1) back to the starting point (0, 0).
The curve forms a non-standard, step-shaped path with clear vertex points at each directional change, as denoted by the grid lines on the plane. The axes are labeled with x and y, and grid lines mark the integers from -2 to 2 on the x-axis and from 0 to 4 on the y-axis.
Note: The orientation of the curve is in a clockwise direction, which is important for evaluating the line integral.
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