Problem 49 - Thomas Calculus

Question

Use a CAS to perform the following steps for the given curve over the closed interval. a. Plot the curve together with the polygonal path approximations for \(n=2,4,8\) partition points over the interval. (See Figure \(11.15 . )\) b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments. c. Evaluate the length of the curve using an integral. Compare your approximations for \(n=2,4,8\) with the actual length given by the integral. How does the actual length compare with the approximations as \(n\) increases? Explain your answer. $$ x=\frac{1}{3} t^{3}, \quad y=\frac{1}{2} t^{2}, \quad 0 \leq t \leq 1 $$

Step-by-Step Solution

Answer

Approximate the curve length using segment sums, then use an integral for the exact length; higher \(n\) yields better approximation.

Step 1: Plot the Curve and Polygonal Paths for n=2, 4, 8

First, use a computer algebra system (CAS) to plot the parametric curve given by the equations \(x=\frac{1}{3}t^3\) and \(y=\frac{1}{2}t^2\) over the interval \(0 \leq t \leq 1\). Then, overlay polygonal path approximations for \(n=2, 4, 8\) partition points within the interval. For each \(n\), evenly distribute the \(t\) values and connect them with straight lines, these lines approximate the curve.

Step 2: Calculate Approximations of Curve Length

For each value of \(n\) (2, 4, 8), calculate the length of the polygonal path by summing the lengths of the individual line segments between partition points. Use the distance formula: \(\text{distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to find the length of each segment, then add them together for the total approximation.

Step 3: Compute Exact Length Using Integral

To find the actual length of the curve, use the arc length formula for parametric equations: \(L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt\). Compute the derivatives: \(\frac{dx}{dt} = t^2\) and \(\frac{dy}{dt} = t\), then set up the integral from \(0\) to \(1\): \[L = \int_0^1 \sqrt{(t^2)^2 + (t)^2} \, dt = \int_0^1 \sqrt{t^4 + t^2} \, dt = \int_0^1 t\sqrt{t^2 + 1} \, dt\]. Evaluate the integral to find the actual curve length.

Step 4: Compare Approximation with Exact Length

Compare the approximate curve lengths for \(n=2, 4, 8\) with the exact length found from the integral. Generally, as \(n\) increases, the approximations should become closer to the actual length. Discuss how the accuracy improves as the number of partition points increases and why the polygonal paths converge to the true curve.

Key Concepts

Parametric Equations

Parametric equations offer a unique way to represent a curve by specifying a set of equations, each expressing a variable as a function of a parameter, commonly denoted as \( t \). This is particularly useful when dealing with more complex curves in geometrical analysis.

• In our exercise, we use the parametric equations \(x = \frac{1}{3}t^3\) and \(y = \frac{1}{2}t^2\) to define a curve over the interval \(0 \leq t \leq 1\).
• Through the parameter \( t \), we control how the curve progresses from its starting point to its ending point.
• The beauty of parametric equations is their ability to model curves that cannot be easily expressed as a single function, \( y = f(x) \), for instance.

By plotting these equations over a specified interval, students can visually perceive the nature of the curve. This method of curve representation gives a broader view into geometry, facilitating various analytical methods like finding derivatives and integrals.

Polygonal Approximation

Polygonal approximation is a method of estimating the length of a curved path by using straight line segments. It’s a simple yet powerful technique often employed when the actual curve length needs estimation, especially when dealing with complex or non-linear curves.

• To approximate the curve defined by the given parametric equations, divide the interval \(0 \leq t \leq 1\) into \(n\) equal parts, where \(n\) could be 2, 4, or 8, according to our task.
• For each partition point, calculate corresponding \(x\) and \(y\) coordinates using the parametric equations.
• Connect these points with straight lines; the collection of these lines forms a polygonal path, offering an estimate of the curve's length.

By increasing \(n\), meaning more line segments are used, the polygonal path becomes a better approximation of the original curve. This method provides a visual and computational insight into understanding curves and calculus, emphasizing the concept that as subdivisions increase, the approximation tends toward the actual figure.

Numerical Integration

Numerical integration is a strategic approach used to evaluate the integral when an analytical solution is challenging or impossible to find outright. This technique is perfect for calculating arc length in scenarios involving complex functions.

• Using the parametric curve's derivatives, \( \frac{dx}{dt} = t^2 \) and \( \frac{dy}{dt} = t \), the arc length integral becomes \( L = \int_0^1 t\sqrt{t^2 + 1} \, dt \).
• When solving this integral, numerical methods such as the trapezoidal rule or Simpson's rule can be very effective if the integral can't be computed easily by hand.
• The integral accounts for every infinitesimally small segment of the curve, summing up these segments to find the total length.

By comparing these computed values with the polygonal approximations, students see how a larger \(n\) leads the polygonal estimate closer to the integral’s calculated value, emphasizing the importance of integral calculus in accurately determining lengths and other properties of curves.