Problem 171 - Calculus Volume 3

Question

Evaluate \(\int_{C}\left(x^{2} y-2 x y+y^{2}\right) d s,\) where \(C\) is the boundary of the unit square \(0 \leq x \leq 1,0 \leq y \leq 1\), traversed counterclockwise.

Step-by-Step Solution

Answer

The integral evaluates to 0 due to symmetry.

Step 1: Understand the Problem

We are asked to evaluate the line integral of the function \( f(x, y) = x^2 y - 2xy + y^2 \) over the curve \( C \) which is the boundary of the unit square in the first quadrant, traversed counterclockwise.

Step 2: Identify the Segments of the Path

The unit square boundary consists of four line segments:\( C_1: (0,0) \to (1,0) \), \( C_2: (1,0) \to (1,1) \), \( C_3: (1,1) \to (0,1) \), and \( C_4: (0,1) \to (0,0) \).

Key Concepts

Unit Square Boundary

The boundary of a unit square is a geometrical concept representing a square with sides of length 1. This particular square is defined in the coordinate plane where each side aligns with either the x-axis or y-axis. The boundaries have specific coordinates starting from the point - (0,0), the square extends to - (1,0) along the x-axis, - (1,1) upwards along the y-axis, - (0,1) back along the x-axis, and finally - (0,0) downwards closing the path.
In the context of a line integral, this closed loop along the unit square's boundary offers a simple but clear path to integrate over. Whenever we discuss the boundary of the unit square, we are referring to these four sides forming the path, specifically defined for this problem.

Counterclockwise Traversal

When tracing the path over the unit square boundary, it is specified that the path should be traversed in a counterclockwise direction. In geometry, counterclockwise traversal means moving in a direction opposite to the way clock hands move. - Beginning from the lower-left corner point (0,0), - the path moves right towards (1,0) - then vertically up to (1,1), - then left towards (0,1), - and finally down back to (0,0).
This order of traversal affects how the line integral is evaluated, as the orientation of the path can change the result of the integral. It is crucial to follow the specified direction to ensure the integral is computed correctly, accurately reflecting the vector field through which the path slices.

First Quadrant

The first quadrant is a part of a Cartesian coordinate system composed of the positive x and y axes. It is one of four quadrants and can be visualized on a graph where both x and y coordinates are greater than zero. This quadrant encompasses any points where - x is between 0 and infinity, and - y is also between 0 and infinity.
For this exercise, the entire operation occurs within the first quadrant. This is especially important since boundary and path segments are always non-negative and within the restricted confines of the first quadrant. Understanding that this exercise lies entirely in the first quadrant helps in visualizing the shape and path of the problem, ensuring accuracy in a real-world graphing context.

Piecewise Path Integration

Piecewise path integration involves breaking down the integral into separate segments, each corresponding to a piece of the underlying path or curve. For the unit square boundary, each side of the square can be treated as a distinct segment. Therefore, the path is divided into: - Segment 1: From (0,0) to (1,0), - Segment 2: From (1,0) to (1,1), - Segment 3: From (1,1) to (0,1), - Segment 4: Back to (0,0).
Each of these segments is integrated separately, and these individual results are summed together to find the complete value of the line integral. This method allows for simplified calculations, responding to each change in direction or function along the path, thereby supporting a clear and structured approach to solving complex integrals along defined paths.