Problem 40 - Thomas Calculus

Question

In Exercises \(35-40,\) use a CAS to perform the following steps for the given graph of the function over the closed interval. \begin{equation}\begin{array}{l}{\text { a. Plot the curve together with the polygonal path approxima- }} \\ {\text { tions for } n=2,4,8 \text { partition points over the interval. (See }} \\ {\text { Figure } 6.22 . )} \\\ {\text { b. Find the corresponding approximation to the length of the }} \\\ {\text { curve by summing the lengths of the line segments. }} \\ {\text { c. Evaluate the length of the curve using an integral. }} \\ {\text { Compare your approximations for } n=2,4,8 \text { with the }} \\ {\text { actual length given by the integral. How does the actual }} \\ {\text { length compare with the approximations as } n \text { increases? }} \\ {\text { Explain your answer. }}\end{array}\end{equation} $$f(x)=x^{3}-x^{2}, \quad-1 \leq x \leq 1$$

Step-by-Step Solution

Answer

The approximated length becomes more accurate as \( n \) increases, approaching the exact integral value.

Step 1: Plot the Curve and Polygonal Path Approximations

Start by plotting the graph of the function \( f(x) = x^3 - x^2 \) over the interval \([-1, 1]\). Using a computer algebra system (CAS), add the polygonal path approximations for partitions \( n=2, 4, \) and \( 8 \). For each \( n \), divide the interval into \( n \) equal segments and draw line segments connecting the function values at these points.

Step 2: Approximate Curve Length with Summed Line Segments

For each value of \( n \), calculate the length of the polygonal path. For \( n=2 \), calculate the length using: \[ L = \sum_{i=1}^{2} \sqrt{(x_{i}-x_{i-1})^2 + (y_{i}-y_{i-1})^2} \] where \((x_i, y_i)\) are the coordinates of the points on the curve. Repeat the same calculation for \( n = 4 \) and \( n = 8 \).

Step 3: Evaluate the Integral for the Exact Length

Use the arc length formula to find the exact length of the curve: \[ L = \int_{-1}^{1} \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx \] First, calculate the derivative of \( f(x) \), \( \frac{df}{dx} = 3x^2 - 2x \). Then, substitute this derivative back into the arc length integral and compute the definite integral in \([-1, 1]\).

Step 4: Compare Approximations with Exact Length

Compare the lengths obtained from polygonal path approximations for \( n=2, 4, \) and \( 8 \) with the exact length found in the previous step. Observe that as \( n \) increases, the length of the polygonal path gets closer to the actual integral length. Discuss how increasing \( n \) improves the approximation by providing a finer partition of the interval, thus better capturing the curve's behavior.

Key Concepts

Polygonal Path Approximation

Imagine trying to walk along a winding path. To understand its length, we can break it down into several, simpler straight paths. This is what we do with polygonal path approximation. It involves representing a curve as a series of straight-line segments. Each segment connects point pairs on the curve, creating a sort of invisible path that shadows the actual curve.

Using a Computer Algebra System (CAS), we plot the function's curve alongside these polygonal paths. By choosing different numbers of segments (like 2, 4, or 8), we can approximate the curve with increased precision. More segments mean a path forged with smaller jumps, adhering more closely to the curve itself.

This visual method provides a tangible way to estimate the curve's length by simply measuring these straight paths.

Partition Points

Partition points are the markers that divide the interval over which the curve is drawn. Imagine breaking a long ribbon into equal little pieces; each cut represents a partition point. For the interval i.e., , we choose how many such cut points (partitions) we want - for instance, dividing the interval into 2, 4, or 8 equal parts.

These points are not arbitrary; they create equal sub-intervals that simplify the calculation of line segments between the curve points. These segments become sides of the polygonal path. The more partition points we use, the closer these segments align to the shape of the curve, leading to a more accurate approximation.

Arc Length Integral

The arc length integral gives us the precise measurement of the curve's length. Instead of summing the lengths of paths, we employ calculus for a more exact computation.

To find this exact arc length, we start with the curve's derivative, which tells us about the curve's slope or direction at any point. For the function , the derivative is . Using this, the formula to compute arc length is:\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx\]
where \(a\) and \(b\) are the interval bounds. Calculating this integral provides the true length by accounting for every minuscule change in the slope along the curve. This is often more challenging to compute but yields the most accurate result.

Curve Approximation

Curve approximation is the method of estimating complex curves using simpler geometric representations. The key is to balance between simplicity and precision, often dictated by available resources or the context of the problem.

Starting with fewer segments might give a rough estimate, serving as a baseline. As we increase the number of segments in the polygonal path approximation, the estimate improves.

• For small \( n \), approximations are quicker but less accurate.
• With larger \( n \), the approximations take more computation but align closer with the curve's true form.

The comparison between these initial estimations and the arc length integral's exact result shows us how approximation limits diminish with refinement. It's a practice seen everywhere in technology: the art of approximating one thing to understand another.