Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 33-36, set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. $$ x=t, y=t^{2}, z=t^{3} ; 1 \leq t \leq 2 $$

Step-by-Step Solution

Answer

The definite integral is set up as \( \int_{1}^{2} \sqrt{1 + 4t^2 + 9t^4} \, dt \) and approximated using Simpson's Rule.

Step 1: Arc Length Formula

To find the arc length of a parametric curve \((x(t), y(t), z(t))\) from \(t = a\) to \(t = b\), we use the formula \( L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt\).

Step 2: Compute Derivatives

Compute the derivatives of each component: \(\frac{dx}{dt} = 1\), \(\frac{dy}{dt} = 2t\), and \(\frac{dz}{dt} = 3t^2\).

Step 3: Set up Integral

Substitute the derivatives into the arc length formula: \[ L = \int_{1}^{2} \sqrt{1^2 + (2t)^2 + (3t^2)^2} \, dt. \] Simplify the expression inside the square root to get: \[ \sqrt{1 + 4t^2 + 9t^4}. \]

Step 4: Approximate the Integral

Use the Parabolic Rule (Simpson's Rule) with \(n = 10\) to approximate the integral. Divide the interval \([1, 2]\) into 10 equal subintervals and apply Simpson's Rule: \[ L \approx \frac{1-2}{3\times 10} \left[f(1) + 4f(1.1) + 2f(1.2) + \ldots + 4f(1.9) + f(2)\right], \] where \( f(t) = \sqrt{1 + 4t^2 + 9t^4}\).

Key Concepts

Parametric Equations

Parametric equations allow us to describe curves in 2D or 3D space using parameters. Unlike traditional equations that use a single input-output relationship, parametric equations utilize one or more variables to describe each axis. This is particularly useful for expressing motion or complex shapes.
For example, in our original exercise, the parametric equations given are:

• \( x = t \)
• \( y = t^2 \)
• \( z = t^3 \)

These equations define a 3D curve where each value of \( t \) determines a specific point \( (x, y, z) \) on the curve. As \( t \) varies from 1 to 2, the curve traces out a path in 3D space. This form makes it easier to handle different shapes and paths, offering flexible modeling of curves that are not easily expressed with traditional Cartesian coordinates.

Definite Integral

A definite integral is a fundamental concept in calculus that computes the accumulation of quantities, like area or, in this case, arc length. In the context of our exercise, it is used to evaluate the total length of a curved path described by the parametric equations.
The formula \[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt \]allows us to calculate the arc length from \( t = a \) to \( t = b \). Each derivative within the formula represents the rate of change of the coordinates with respect to the parameter \( t \).
By solving this integral, we understand not only the simple geometry of the path, but also how it changes and evolves, which is essential for fields like physics and engineering where precise shapes are critical.

Simpson's Rule

Simpson's Rule is a numerical method used to approximate definite integrals. It's particularly advantageous when the integral is difficult or impossible to evaluate analytically. In the context of the given exercise, Simpson's Rule provides a way to estimate the arc length of the curve using computational techniques.
Simpson's Rule works by fitting a parabola through each pair of subintervals and using these parabolas to estimate the area under the curve. It uses the formula:
\[ L \approx \frac{b-a}{3n} \left[f(a) + 4f(a+h) + 2f(a+2h) + \ldots + 4f(b-h) + f(b)\right] \]where \( h = \frac{b-a}{n} \) and \( n \) is an even number of intervals. In practice, this means we are using points along the curve to create small approximations that add up to a close estimate of the total integral.

• Accurate for smooth functions
• Easy to implement with calculators or computer software
• Useful for approximating integrals over finite intervals

By using Simpson's Rule, we can achieve close approximations of complex curves, making it practical for real-world applications and scenarios where exact analytical solutions are unavailable.