Problem 38 - 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} $$f(x)=x^{2} \cos x, \quad 0 \leq x \leq \pi$$

Step-by-Step Solution

Answer

The curve length approximation improves as \( n \) increases, matching the integral length closer.

Step 1: Understand the Function and Interval

We are given the function \( f(x) = x^2 \cos x \) and the interval \( [0, \pi] \). The goal is to approximate the length of the curve over this interval using both a polygonal path approximation with \( n=2, 4, \) and \( 8 \) partitions and the actual integral.

Step 2: Plot the Curve and Approximate Path

Using a Computer Algebra System (CAS), plot the function \( f(x) = x^2 \cos x \) from \( x=0 \) to \( x=\pi \). Generate polygonal path approximations by dividing the interval into \( n=2, 4, \) and \( 8 \) equal subintervals. Draw line segments connecting the function values at these partition points to approximate the curve.

Step 3: Calculate Length of Polygonal Path for n=2

Divide the interval \([0, \pi]\) into \( n=2 \) equal parts and calculate the length of each line segment using the distance formula. Sum these lengths to approximate the length of the curve.

Step 4: Calculate Length of Polygonal Path for n=4

Subdivide the interval into \( n=4 \) parts and repeat the process from Step 3 to find and sum the lengths of the line segments for the new partition.

Step 5: Calculate Length of Polygonal Path for n=8

Further refine the interval into \( n=8 \) parts and calculate the length of each segment. Sum these lengths for the \( n=8 \) partition approximation.

Step 6: Evaluate the Actual Curve Length Using Integral

Set up the integral \( L = \int_0^\pi \sqrt{1 + (\frac{d}{dx}(x^2 \cos x))^2} \ dx \) to find the actual length of the curve. Use your CAS to evaluate this integral. The derivative \( \frac{d}{dx}(x^2 \cos x) \) is \( 2x \cos x - x^2 \sin x \).

Step 7: Compare the Approximations to the Actual Length

Compare the approximated lengths from Steps 3, 4, and 5 with the actual length from Step 6. Observe that as \( n \) increases, the polygonal approximation converges to the actual curve length. The approximations become more accurate with increased \( n \) due to more segments better capturing the curvature of \( f(x) \).

Key Concepts

Curve Length

The concept of curve length in calculus addresses finding the distance along a curve between two points. Unlike straight lines, curves require careful approaches to determine their length because they twist and turn in space.
To calculate the exact length of a curve described by a function, we use integral calculus. The formula for the length of a curve defined by the function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by the integral \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \).
This formula considers not only the distance in the \( x \)-direction but also the changes in the \( y \)-direction, providing a comprehensive measure of the curve's length.

Polygonal Path Approximation

A polygonal path approximation helps to estimate the length of a curve by replacing it with straight-line segments. In this method, the curve's interval is divided into several subintervals, and a line segment approximates the curve within each interval.
By increasing the number of segments \( n \), we can improve the approximation. The more segments used, the closer the approximation is to the true curve. This is because each line segment better captures the slope and bend of the original curve.

• For \( n=2 \), the approximation is rough, likely underestimating the curve.
• With \( n=4 \) or \( n=8 \), the approximation becomes finer, reducing the error.

Such approximations are foundational in numerical methods and serve as a stepping stone to understanding definite integrals.

Definite Integral

Definite integrals are critical in calculus and serve as a powerful tool for finding areas, volumes, and in this case, the length of curves. When computing the curve length, the integral \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \) is evaluated.
The definite integral sums up infinitely many infinitesimally small pieces along the curve, giving an exact value for the length. It considers the changes in heights and widths between each infinitesimal interval.
In comparing polygonal path approximations with the curve length from a definite integral, we notice that the definite integral provides a precise result, while the approximations average towards this result as \( n \) increases.

Function Plotting

Function plotting is essential in visualizing mathematical relationships, especially in calculus where functions can behave in complex ways. For the function \( f(x) = x^2 \cos x \), plotting helps you see how the function behaves over a specific interval, such as \([0, \pi]\).
By using a computer algebra system (CAS), you can easily generate graphs and explore both the function and its approximations. This visual insight provides clues about how the points are connected, the steepness of the curve, and where approximations might over or under estimate.

• Ensure your graph includes both the actual curve and the polygonal approximation for various \( n \).
• Observing changes in the graph helps verify both visual and numerical accuracy of your curve approximations.

Function plotting reinforces our understanding by integrating visual and computational analyses.