Calculating the Arc Length of a Parametric Curve: A Comprehensive Guide

In calculus, calculating the arc length of a parametric curve is a fundamental concept. Parametric curves are defined by a pair of equations where one variable depends on another. In this article, we will delve into how to find the arc length of a parametric curve given by:

Given Equations

The parametric equations given are:

x'(t) frac{t}{1 - t^2} y'(t) frac{1}{1 - t^2}

The requested length L can be found using the following integral:

L int_{0}^{1} sqrt{(x'(t)^2 y'(t)^2)} , dt

Substituting the given parametric equations into the integral, we get:

L int_{0}^{1} sqrt{left(frac{t}{1 - t^2}right)^2 left(frac{1}{1 - t^2}right)^2} , dt

Simplifying the integrand, we have:

L int_{0}^{1} frac{1}{sqrt{1 - t^2}} , dt

Integration Using Trigonometric Substitution

To solve this integral, we use the substitution t tan theta. From this, we get:

dt sec^2 theta , dtheta

Also, sec^2 theta 1 tan^2 theta 1 t^2. Substituting this into the integral, we get:

L int_{0}^{frac{pi}{4}} sec theta , dtheta

Now, integrating:

L left[ln (sec theta tan theta)right]_{0}^{frac{pi}{4}}

Evaluating the limits, we obtain:

L ln (sec frac{pi}{4} tan frac{pi}{4}) - ln (sec 0 tan 0)

Solving the above expression, we get:

L ln (sqrt{2} 1) - ln (1 0) ln left(sqrt{2} 1right)

Thus, the arc length of the curve is:

L ln left(frac{1}{sqrt{2} - 1}right)

This can be simplified further to:

L ln (sqrt{2} 1)

Alternative Solution

Using an alternative method, let's consider the parametric equations:

x frac{1}{2} ln (t^2 - 1) frac{dx}{dt} frac{t}{t^2 - 1} frac{dy}{dt} frac{1}{t^2 - 1}

The arc length L is given by:

L int_{0}^{1} sqrt{left(frac{dx}{dt}right)^2 left(frac{dy}{dt}right)^2} , dt

Substituting the derivatives:

L int_{0}^{1} sqrt{left(frac{t}{t^2 - 1}right)^2 left(frac{1}{t^2 - 1}right)^2} , dt

Simplifying the integrand:

L int_{0}^{1} frac{1}{sqrt{t^2 - 1}} , dt

Using the substitution sec^2 theta , dtheta dt, we have:

L int_{0}^{frac{pi}{4}} sec theta , dtheta

Integrating:

L left[ln (sec theta tan theta)right]_{0}^{frac{pi}{4}}

Evaluating the limits, we obtain:

L ln (sec frac{pi}{4} tan frac{pi}{4}) - ln (sec 0 tan 0)

Solving the above expression, we get:

L ln (sqrt{2} 1) - ln (1 0) ln left(sqrt{2} 1right)

Thus, the arc length of the curve is:

L ln (sqrt{2} 1)

Conceptual Summary

The arc length of a parametric curve is an important concept in calculus that involves integrating the square root of the sum of the squares of the derivatives of the parametric equations. This can be computed using various methods including trigonometric substitutions or direct integration.

The integral approach to finding the arc length is useful in a wide range of applications, from physics and engineering to advanced mathematical models. Understanding the techniques and methods used to solve these types of integrals is crucial for mathematicians, engineers, and scientists.

Conclusion

In conclusion, calculating the arc length of a parametric curve involves a series of steps, including the application of derivatives and integration techniques. The process not only helps in understanding the geometric properties of curves but also provides a basis for more complex mathematical modeling and analysis.

For further exploration, consider practicing more problems and exploring different parametric equations. Understanding the underlying principles and methods will aid in tackling a broader range of problems.