Question

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. \(x=t, y=t^{2}, z=t^{3} ; 1 \leq t \leq 2\)

Short answer

Approximate arc length using numerical methods: \( L \approx \int_1^2 \sqrt{1 + 4t^2 + 9t^4} \, dt \).

Step-by-step solution

Understand the Arc Length Formula

For a parametric curve given by \(x(t), y(t), z(t)\), the arc length from \(t=a\) to \(t=b\) is given by the integral \(L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt\). Our task is to calculate this integral for the provided parametric equations.

Differentiate the Curve with Respect to t

Find the derivatives: \(\frac{dx}{dt} = 1\), \(\frac{dy}{dt} = 2t\), and \(\frac{dz}{dt} = 3t^2\). These derivatives represent the rates of change of the x, y, and z components with respect to the parameter \(t\).

Substitute the Derivatives into the Arc Length Formula

Plug in the derivatives into the arc length formula: \[ L = \int_1^2 \sqrt{1^2 + (2t)^2 + (3t^2)^2} \, dt \]This simplifies to \[ L = \int_1^2 \sqrt{1 + 4t^2 + 9t^4} \, dt \].

Approximate the Integral Using the Parabolic Rule

Use numerical methods, either the Parabolic rule (also known as Simpson's rule) or a Computer Algebra System (CAS), to approximate this definite integral. With \(n=10\), you divide the interval \([1, 2]\) into 10 sub-intervals and compute the approximate value for the integral \[ L \approx \int_1^2 \sqrt{1 + 4t^2 + 9t^4} \, dt \] using the rule. Note that a detailed manual calculation is made feasible with software or a calculator that supports such integration.

Key concepts

Understanding Parametric Equations

Parametric equations are mathematical expressions where a curve is represented by a parameter, usually denoted as \( t \). Unlike conventional equations in the form \( y = f(x) \), parametric equations define both \( x \), \( y \), and sometimes \( z \) as separate functions of \( t \). This allows for a more flexible description of curves that can express more complex paths in the plane or in space.

In the given exercise, we have three parametric equations: \( x = t \), \( y = t^2 \), and \( z = t^3 \). Here, \( t \) varies from 1 to 2, generating a curve in three-dimensional space. Parametric equations are especially handy when dealing with curves that fold back on themselves or ones where a single \( y \) for each \( x \) does not exist.

The beauty of parametric equations is their ability to simplify calculations for curves by breaking them into component parts, which is crucial when solving problems like finding the arc length.

Numerical Integration Simplified

Numerical integration is a computational approach used when finding the exact analytic solution of an integral is challenging or impossible. It involves approximating the integral by summing the values of a function at specific points within the interval over which the integral is evaluated. This technique becomes essential in calculating definite integrals where traditional methods don't provide a clear solution.

In the exercise, the arc length integral we need to solve is expressed as \( \int_1^2 \sqrt{1 + 4t^2 + 9t^4} \, dt \). Such an integral might not have a simple antiderivative, making numerical integration methods like Simpson's Rule or using a Computer Algebra System (CAS) invaluable.

Methods for numerical integration range from simple Riemann sums to more advanced techniques like the trapezoidal rule and Simpson's Rule, each balancing complexity and approximation accuracy. The goal is to approximate the integral with sufficient precision for practical purposes.

Exploring Derivatives in Calculus

In calculus, derivatives measure how a function changes as its input changes. Understanding derivatives is crucial for determining the behavior of parametric equations and, consequently, the calculation of curve properties such as arc length.

For the parametric equations \( x = t \), \( y = t^2 \), and \( z = t^3 \), we find their respective derivatives: \( \frac{dx}{dt} = 1 \), \( \frac{dy}{dt} = 2t \), and \( \frac{dz}{dt} = 3t^2 \). These derivatives denote the rate of change of each coordinate with respect to the parameter \( t \).

Calculating these derivatives allows us to substitute into the arc length formula \( L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2} \, dt \). This step ensures we accurately capture how each dimension contributes to the total length of the curve, providing a more comprehensive understanding of its geometrical properties.

Simpson's Rule for Approximation

Simpson's Rule is a numerical method used specifically for approximating the definite integral of a function. It is an upgrade from the trapezoidal rule and achieves higher accuracy by considering parabolic segments instead of straight-line segments.

This rule approximates the area under a curve by dividing it into an even number \( n \) of sub-intervals. Each pair of sub-intervals is used to form a parabola that approximates the area more closely than straight lines. The formula for Simpson's Rule is:

• \( \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n) \right] \)

where \( \Delta x \) is the width of each sub-interval and \( f \) is the function being integrated.

When applied to our exercise, Simpson's Rule allows us to approximate the arc length integral \( \int_1^2 \sqrt{1 + 4t^2 + 9t^4} \, dt \) with an appropriate number of divisions (in this case \( n=10 \), chosen for better precision) and provides a more practical solution than attempting analytical integration.