Problem 14 - Advanced Engineering Mathematics

Question

Evaluate \(\int_{C} y d x+x d y\) on the given curve \(C\) between \((0,0)\) and \((1,1)\). \(C\) consists of the line segments from \((0,0)\) to \((1,0)\) and from \((1,0)\) to \((1,1)\).

Step-by-Step Solution

Answer

The integral evaluates to 1.

Step 1: Parametrize the First Curve Segment

The first segment of the curve goes from \((0,0)\) to \((1,0)\), which is a horizontal line along the x-axis. We can parametrize it using \(x = t\) and \(y = 0\), where \(t\) ranges from 0 to 1.

Step 2: Substitute and Evaluate the Integral on the First Segment

Substitute the parametrization into the integral: \(\int_{C_1} y \, dx + x \, dy = \int_{0}^{1} 0 \, dt + \int_{0}^{1} t \, d(0) = 0\). As \(y = 0\), the expression reduces to 0 because both contributions vanish.

Step 3: Parametrize the Second Curve Segment

The second segment of the curve goes from \((1,0)\) to \((1,1)\), which is a vertical line. We can parametrize it using \(x = 1\) and \(y = t\), where \(t\) ranges from 0 to 1.

Step 4: Substitute and Evaluate the Integral on the Second Segment

Substitute the new parametrization into the integral: \(\int_{C_2} y \, dx + x \, dy = \int_{0}^{1} t \, d(1) + \int_{0}^{1} 1 \, dt\). Since \(dx = 0\), the first term vanishes, so it becomes \(\int_{0}^{1} 1 \, dt = [t]_{0}^{1} = 1\).

Step 5: Combine Results From Both Segments

The total integral along the curve \(C\) is the sum of the integrals from the two segments: \(0 + 1 = 1\). Thus, the integral evaluated over the entire curve \(C\) is 1.

Key Concepts

Parametrization of Curves

Parametrization of curves is a powerful technique in vector calculus that allows us to express a curve as a set of equations. This is essential for evaluating integrals along a curve, particularly when the curve is piecewise or segmented.

To parametrize a curve, you assign a parameter, often denoted as \(t\), to the coordinates of the curve. Let's consider our case:

• First Segment: The line from \((0,0)\) to \((1,0)\) is horizontal. We choose \(x = t\) and \(y = 0\), where \(t\) varies from 0 to 1.
• Second Segment: The line from \((1,0)\) to \((1,1)\) is vertical. We pick \(x = 1\) and \(y = t\), with \(t\) ranging from 0 to 1.

With these parametrizations, we can easily substitute values into integrals. This method transforms what might look like a complex curve into manageable linear segments.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and differentiable fields. It extends calculus into higher dimensions, providing tools for evaluating integrals over different paths and surfaces.

In this exercise, the line integral involves differential elements \(dx\) and \(dy\) that affect how we compute integrals depending on the direction and path:

• Vector Function: We consider the vector field given by \(\mathbf{F} = (y, x)\). Evaluating these components over a parametrization helps in calculating integrals.
• Differential Path Elements: Depending on the parametrization, \(dx\) and \(dy\) take specific forms that correspond to the change along \(x\) and \(y\) as you move along the curve.

These principles permit us to calculate line integrals, a central concept in physics and engineering, representing work done along paths or other physical interpretations.

Evaluating Integrals on Piecewise Paths

Evaluating integrals on piecewise paths involves splitting the integral into parts that correspond to individual segments of a curve. These path segments often have different mathematical expressions, and managing them separately eases computation.

Here is how the process works:

• Step 1: Calculate the integral for each piece. For our first segment from \((0,0)\) to \((1,0)\), after substitution, the integral evaluates to zero because both terms vanish.
• Step 2: For the second segment from \((1,0)\) to \((1,1)\), after substitution, the integral yields a result of 1.
• Step 3: Combine results from each segment to find the total value. This sum gives us the integral over the entire curve.

This piecewise approach is practical for complex curves broken into linear or manageable segments, allowing each segment's contributions to be simple to calculate and sum.