How to Calculate the Line Integral Using Parameterization

Introduction

Line integrals are a fundamental concept in vector calculus, often used to calculate the work done by a force along a curve. In this article, we will demonstrate how to calculate a line integral using parameterization. Specifically, we will use the parametric equations (x t), (y t), and (z t) for (t in [0, a]) to express a line integral as a standard definite integral.

Parameterizing the Curve

Given a curve (C) defined by the parametric equations (x t), (y t), and (z t) for (t in [0, a]), we can express the vector field (mathbf{f}) as a function of these parameters. Let's consider a specific example where (mathbf{f} frac{-2}{3t^2 - a^2} langle t, t, t rangle cdot langle 1, 1, 1 rangle).

Step 1: Setting Up the Definite Integral

First, we rewrite the line integral (int_C mathbf{f} cdot dmathbf{r}) as a standard definite integral:

[int_C mathbf{f} cdot dmathbf{r} int_0^a frac{-2}{3t^2 - a^2} langle t, t, t rangle cdot langle 1, 1, 1 rangle , dt int_0^a frac{-6t}{3t^2 - a^2} , dt.] (1)

Step 2: Solving the Definite Integral

The integral on the right-hand side of equation (1) can be evaluated by using a substitution. Let (w 3t^2 - a^2). Then, we have:

[dw 6t , dt.] (2)

When (t 0), (w -a^2). And when (t a), (w 2a^2).

The integral becomes:

[int_0^a frac{-6t}{3t^2 - a^2} , dt int_{-a^2}^{2a^2} frac{-1}{w} , dw.] (3)

Integrating, we find:

[int_{-a^2}^{2a^2} frac{-1}{w} , dw -ln(w) |_{-a^2}^{2a^2}.] (4)

Evaluating the limits, we get:

[-ln(2a^2) - (-ln(-a^2)) -ln(2a^2) ln(-a^2) -ln(4).] (5)

Conclusion

Therefore, the line integral (int_C mathbf{f} cdot dmathbf{r}) evaluates to (-ln(4)) using parameterization and definite integral techniques.

1. Calculus, Vol. 2, by Tom M. Apostol, Chapter 14, Section 14.1, Example 2

2. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, by John H. Hubbard and Barbara Burke Hubbard, Chapter 7, Section 7.3, Example 7.3.2

3. Vector Calculus, by Jerrold E. Marsden and Anthony Tromba, Chapter 5, Section 5.2, Example 5.2.4