Problem 35 - 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} $$f(x)=\sqrt{1-x^{2}},-1 \leq x \leq 1$$

Step-by-Step Solution

Answer

The actual length of the curve is \( \pi \). Approximations improve as \( n \) increases, approaching \( \pi \).

Step 1: Plotting the Curve and Approximations

First, plot the curve of the function \( f(x) = \sqrt{1-x^2} \) over the interval \([-1, 1]\). This curve is the upper half of the unit circle. For each partition with \( n = 2, 4, 8 \), divide the interval into equal parts, and draw line segments between the partition points to approximate the curve.

Step 2: Calculating the Approximation for Curve Lengths

For \( n=2 \), the interval \([-1, 1]\) is divided into two parts, giving partition points \(-1, 0, 1\). Calculate the lengths of the segments connecting these points using the distance formula. Repeat this for \( n=4 \) and \( n=8 \), where the interval is divided into 4 and 8 segments, respectively. Sum up the lengths of the line segments to get the total approximate length of the curve for each \( n \).

Step 3: Exact Curve Length via Integration

The actual length of the curve can be determined using the integral \[L = \int_{-1}^1 \sqrt{1 + \left(\frac{df}{dx}\right)^2} \, dx\]Since \( f(x) = \sqrt{1-x^2} \), differentiate to find \( \frac{df}{dx} = \frac{-x}{\sqrt{1-x^2}} \). Then, compute:\[L = \int_{-1}^1 \frac{1}{\sqrt{1-x^2}} \, dx\]This evaluates to \( \pi \), which is the half circumference of the unit circle.

Step 4: Comparison and Summary

Now, compare the lengths obtained from approximations (from Step 2) with the exact length from the integral (Step 3). As \( n \) increases, the approximation becomes closer to the actual length, \( \pi \). The approximations improve because increasing \( n \) provides a better piecewise linear approximation to the actual curve. This demonstrates that as the number of segments increases, the polygonal path better represents the curve's actual length.

Key Concepts

Numerical Integration

Numerical integration is a way to approximate the value of an integral when an exact solution is difficult or impossible to find. In many cases, such as with complex functions or when a function does not have an antiderivative, numerical methods can be useful.

• It involves computing the sum of areas under a curve represented by shapes like rectangles, trapezoids, or other polygons.
• The method produces a numerical estimate of the integral by approximating the function with simpler, calculable shapes.

For example, by increasing the number of subdivisions (or partitions, noted by \( n \) in the exercise), the accuracy of the numerical integration improves, since smaller segments can better follow the curve of the function. This method is especially beneficial when paired with computational tools, which can handle the large calculations efficiently.

Polygonal Approximation

Polygonal approximation is a technique used to approximate the length or shape of a curve by connecting a series of points along the curve with straight line segments. This method is quite literally drawing a polygonal path that follows the main features of the curve.

• In the given exercise, partitioning the interval into segments such as \( n = 2, 4, 8 \) helps to create these line segments.
• More partition points result in more segments and a closer approximation to the actual curve.

Think of polygonal approximation as a way of replacing curved sections with small straight lines that mimic the curve. As more line segments are used, the difference between the length of the polygonal path and the actual curve length decreases. The key here is the balance between computational efficiency and accuracy.

Distance Formula

The distance formula is fundamental in calculating the length of the line segments created in a polygonal approximation. When we have two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between these points is given by the formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

• Each segment of the polygonal approximation uses this formula to determine its length.
• For more accurate results, more segments (more distance calculations) mean the sum of line segment lengths approaches the true curve length.

Using the distance formula allows us to quantitatively approach the task of measuring curve lengths, essential in applying polygonal approximations to problems in physics and engineering.

Integral Calculus

Integral calculus is the branch of mathematics concerned with finding integrals or antiderivatives. It allows us to calculate areas under curves, volumes, and in this case, the exact length of curves. It is the inverse process of differentiation.

• The fundamental theorem of calculus links the concept of differentiation with integration, showing that integration is, in a sense, the reverse of finding a derivative.
• In the exercise, the length of the curve \(f(x) = \sqrt{1-x^2}\) is found using the integral \(L = \int_{-1}^1 \frac{1}{\sqrt{1-x^2}} \, dx\), which results in \( \pi \).

Integral calculus is powerful because it provides precise answers unlike approximation methods. By comparing these exact results with numerical approximations, students can understand the reliability and validity of their numerical methods.