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. \(x=2 \cos t, y=\sin t, z=t ; 0 \leq t \leq 6 \pi\)

Short answer

The approximate arc length is found using Simpson's Rule for the integral of the arc length formula.

Step-by-step solution

Arc Length Formula for Parametric Curves

For a curve defined parametrically by \( x = f(t), y = g(t), z = h(t) \), the arc length \( L \) from \( t=a \) to \( t=b \) is given by the integral:\[L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt\]Here, \( a = 0 \) and \( b = 6\pi \).

Find Derivatives

Calculate the derivatives:- \( \frac{dx}{dt} = \frac{d}{dt}(2 \cos t) = -2 \sin t \).- \( \frac{dy}{dt} = \frac{d}{dt}(\sin t) = \cos t \).- \( \frac{dz}{dt} = \frac{d}{dt}(t) = 1 \).

Substitute and Simplify

Substitute the derivatives into the arc length formula:\[L = \int_{0}^{6\pi} \sqrt{(-2 \sin t)^2 + (\cos t)^2 + (1)^2} \, dt\]Simplify the expression under the square root:\[= \int_{0}^{6\pi} \sqrt{4 \sin^2 t + \cos^2 t + 1} \, dt\]\[= \int_{0}^{6\pi} \sqrt{4 \sin^2 t + \cos^2 t + 1} \, dt\]

Simplify Further (Optional)

Notice that under the square root the expression can be rearranged:\[4 \sin^2 t + \cos^2 t + 1 = 1 + \cos^2 t + 3(\sin^2 t + \cos^2 t) = 1 + 3(1) = 4\]This doesn't greatly simplify further with elementary functions because of the trigonometric identity, so we'll integrate directly or use a numerical method.

Approximating the Integral with Parabolic Rule

We use the Parabolic Rule, also known as Simpson's Rule, which is given by:\[\int_{a}^{b} f(t) \, dt \approx \frac{h}{3} \left[ f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + \cdots + 4f(t_{n-1}) + f(t_n) \right]\]where \( h = \frac{b-a}{n} \) and \( n = 10 \) in this case.

Calculate Numerical Approximation

Calculate using Simpson's Rule with the given function under the integral:- Subdivide the interval from \( 0 \) to \( 6\pi \) into 10 sub-intervals.- Use each sub-interval endpoint to calculate the function under the integral and apply the weights as required.- Sum the contributions to get an approximate value for \( L \).

Key concepts

Parametric Curves

Parametric curves allow us to describe a curve by defining both x and y (and possibly z) as functions of a parameter, usually denoted as t. This method is especially useful for complex curves where single equations like y = f(x) might be cumbersome or even impossible.

To imagine this, think of drawing a curve by tracing a path in space. The parameter t essentially traces out this path over a given interval, creating a flexible description that can represent any type of curve, in 2D or 3D. In our original exercise, we have:

• x = 2 cos t
• y = sin t
• z = t

Together, these describe a helix over the interval from 0 to 6π. Each variable being a function of t allows us to approach the problem of finding arc length with clarity and precision.

Definite Integral

A definite integral is a fundamental concept in calculus used to find the exact value of the area under a curve between two specified limits. The integral not only helps in calculating areas but also in determining lengths among other applications.

In the context of parametric curves, the definite integral calculates the length of a curve between two points. The arc length L, as seen in the solution, is specifically defined from one point to another using an integral:\[L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt\]This formula captures the essence of the curve's length by considering the changes along x, y, and z as the parameter t changes from a to b.

In simpler terms, it's like a measure of the distance one travels along the curve from the start (\(t = 0\) in our example) to the end (\(t = 6\pi\)).

Numerical Approximation

Numerical approximation comes into play whenever it's challenging to solve an integral exactly using analytical methods. Techniques like the Parabolic Rule help in estimating the value of an integral when the function is too complex for standard calculus methods.

This includes breaking down the integral into manageable segments that can be calculated and summed to provide a close approximation to the precise value. Using these methods is particularly useful when the exact evaluation of an integral is difficult or impossible.

• Provides a practical solution to complex problems.
• Allows the approximation of integrals for real-world applications where exact precision isn't feasible.

Calculating arc lengths for parametric curves often necessitates the use of such numerical methods as pure algebraic simplification may not always be attainable. In this case, numerical approximation is key to solving the original exercise.

Simpson's Rule

Simpson's Rule is a method of numerical integration used to approximate definite integrals. It's part of a class of techniques that rely on dividing the integral into smaller parts and then calculating the approximate area of each.

Simpson's Rule is particularly accurate when the function is well-behaved, meaning it doesn't have sharp bends or discontinuities. For the rule, the interval is divided into an even number of sub-intervals (n must be even). The endpoints and midpoints of these intervals are used to calculate a weighted average:

• \[ \int_{a}^{b} f(t) \, dt \approx \frac{h}{3} \left[ f(t_0) + 4f(t_1) + 2f(t_2) + \ldots + 4f(t_{n-1}) + f(t_n) \right] \]
• Here, \( h = \frac{b-a}{n} \), and each term's weight corresponds to its position within the interval.

By applying Simpson's Rule to the function found in our arc length calculation, we subdivide the curve into segments, calculate their estimated lengths, and sum them to approximate the integral. This blend of algebra and arithmetic makes Simpson’s Rule both practical and efficient for many applications, including the original problem.