Problem 51 - 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. $$ x=t-\cos t, \quad y=1+\sin t, \quad-\pi \leq t \leq \pi $$

Step-by-Step Solution

Answer

As \(n\) increases, the approximation improves, closely matching the integral's result, showing convergence to the actual curve length.

Step 1: Graph the Curve

Use a computer algebra system (CAS) to plot the given curve described by the parametric equations \( x = t - \cos t \) and \( y = 1 + \sin t \) over the interval \(-\pi \leq t \leq \pi \). This forms a cycloidal curve.

Step 2: Plot Polygonal Path for n=2

Partition the interval \([-\pi, \pi]\) into 2 equal sub-intervals. Calculate the coordinate points corresponding to these partition points, and use them to create a polygonal path approximation of the curve. Plot this alongside the curve.

Step 3: Plot Polygonal Path for n=4

Partition the interval \([-\pi, \pi]\) into 4 equal sub-intervals. Calculate the coordinate points at these partition points and create a polygonal path approximation with these points, plotting them with the curve.

Step 4: Plot Polygonal Path for n=8

Partition the interval \([-\pi, \pi]\) into 8 equal sub-intervals. Determine the coordinates for each partition point and use them to sketch a polygonal path approximation, plotting it together with the curve.

Step 5: Compute Length Approximations for n=2

Calculate the segments' lengths of the polygonal path for \(n=2\) using the distance formula and sum these lengths to approximate the curve length.

Step 6: Compute Length Approximations for n=4

For \(n=4\), compute each line segment's length using the coordinates obtained, sum them up, and get the approximate length of the curve.

Step 7: Compute Length Approximations for n=8

With \(n=8\), calculate the length of each segment using the same method, add these lengths to approximate the curve's total length.

Step 8: Evaluate the Integral for Actual Length

Evaluate the integral to find the actual length of the curve using the formula for arc length in parametric form: \( L = \int_{-\pi}^{\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \). Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\), substitute into the formula, and evaluate.

Step 9: Compare Approximations to Actual Length

Compare the approximated lengths for \(n=2, 4, 8\) with the integral's result. Note how the approximation improves and becomes closer to the actual length as \(n\) increases. Discuss the typical behavior where more partition points generally yield a more accurate approximation.

Key Concepts

Arc Length

When discussing a curve's arc length, imagine measuring the path you would walk if the curve were a road. In this context, our curve is defined by parametric equations. These are equations where each coordinate (x and y) is expressed in terms of a third variable, commonly "t". For the given curve, the coordinates are given by: - \( x = t - \cos t \)- \( y = 1 + \sin t \)Arc length can be found using the formula specifically for parametric equations:\[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]Here, \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) are the derivatives of the parametric equations with respect to \( t \). The limits \( a \) and \( b \) represent the interval over which you want to find the length—in this case, \(-\pi\) to \(\pi\). By finding these derivatives, substituting them into the formula, and evaluating the integral, you obtain the actual arc length of the given curve. The challenge in exercises like this lies in manipulating the derivatives and evaluating the integral accurately.

Polygonal Approximation

To intuitively understand a curve's length, polygonal approximation provides a practical approach. It involves breaking the curve into simpler, straight-line segments. This is akin to forming a polygon by connecting a series of points along the curve.In our exercise, this method begins by choosing a number \( n \) of partition points along the curve's interval \([-\pi, \pi]\). With these points calculated, lines are drawn connecting consecutive points:- For \( n=2 \), the interval is split into two segments, forming a basic polygonal path.- As \( n \) increases to 4 or 8, the approximation becomes finer, more segments are used, and the path begins to mirror more closely the actual curve shape.The length of these line segments is added up to create an approximate total curve length. Of course, this estimate improves with more segments, making it a useful method to intuitive grasp arc length before tackling the exact integral.

Integral Evaluation

Evaluating integrals in arc length problems involves detailed calculus. First, derivatives of the parametric equations \( x(t) \) and \( y(t) \) must be calculated. Specifically, \( \frac{dx}{dt} = 1 + \sin t \) and \( \frac{dy}{dt} = \cos t \).Substituting these into the arc length formula:\[ L = \int_{-\pi}^{\pi} \sqrt{(1 + \sin t)^2 + (\cos t)^2} \, dt \]requires you to simplify and evaluate this expression under the integral sign. Often, simplifications (like using identities \( \sin^2 t + \cos^2 t = 1 \)) help reduce complexity.Using numerical or symbolic computation tools is common to manage intricate integration, ensuring precision where manual calculation may be cumbersome. After evaluating, you have the "true" arc length, an essential benchmark when comparing against the approximations from the polygonal methods.