Problem 50 - 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. $$ x=2 t^{3}-16 t^{2}+25 t+5, \quad y=t^{2}+t-3, \quad 0 \leq t \leq 6 $$

Step-by-Step Solution

Answer

The actual length is more accurate as \(n\) increases, matching the integral result.

Step 1: Set Up the Function

Define the parametric functions for the given curve: \( x(t) = 2t^3 - 16t^2 + 25t + 5 \) and \( y(t) = t^2 + t - 3 \). The interval is \( 0 \leq t \leq 6 \).

Step 2: Plot the Curve with Polygonal Approximations

Use a computer algebra system (CAS) to plot the curve defined by \( x(t) \) and \( y(t) \) along with polygonal path approximations using partition points for \( n=2,4,8 \). The curve should be plotted on the domain \( t \in [0,6] \). Each approximation should consist of line segments connecting the partition points on the curve.

Step 3: Calculate Polygonal Path Lengths

For each \( n=2, 4, 8 \), calculate the polygonal path length. This involves summing up the Euclidean distances between consecutive partition points \((x(t_i), y(t_i))\) where \( t_i = i \cdot \frac{6}{n} \) for \( i = 0, 1, ..., n \). Use the distance formula: \( \sqrt{(x(t_{i+1}) - x(t_i))^2 + (y(t_{i+1}) - y(t_i))^2} \). Sum these distances to get the total length for each approximation.

Step 4: Evaluate the Curve Length Using an Integral

Calculate the actual length of the curve using the arc length formula: \[ L = \int_{0}^{6} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]Find the derivatives \( \frac{dx}{dt} = 6t^2 - 32t + 25 \) and \( \frac{dy}{dt} = 2t + 1 \). Substitute these into the arc length formula and evaluate the integral to find the exact length of the curve.

Step 5: Compare Approximations to the Actual Length

Compare the polygonal path lengths for \( n=2,4,8 \) with the integral's result. Observe how the approximation improves and becomes more accurate as \( n \) increases. Discuss how as \( n \) increases, the summed lengths of the polygonal paths should approach the actual length calculated via the integral. This demonstrates the convergence of Riemann sums to definite integrals.

Key Concepts

Parametric Equations

Parametric equations offer a way to describe curves in the plane. Instead of representing a curve with a single equation, parametric equations use a pair of equations to express both coordinates, usually in terms of a third variable known as the parameter. In the exercise, we see this with our parameter being \( t \).
For the provided curve, the equations are:

• \( x(t) = 2t^3 - 16t^2 + 25t + 5 \)
• \( y(t) = t^2 + t - 3 \)

As \( t \) changes, it dictates the position \((x(t), y(t))\) of a point moving along the curve. The interval \(0 \leq t \leq 6\) specifies the section of the curve we are interested in. This representation is particularly valuable for describing more complex paths and lends itself nicely to calculations using derivatives, making it ideal for calculus applications.

Arc Length

Arc length is about measuring the distance along a curve. It's similar to the way you measure the straight-line distance between two points, except now you do it along a curved path.
In the case of parametric equations, the arc length over an interval \([a, b]\) can be found using the integral formula:\[L = \int_{a}^{b} \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt\]This formula takes into account the changing slope along the curve by using derivatives:

• \( \frac{dx}{dt} = 6t^2 - 32t + 25 \)
• \( \frac{dy}{dt} = 2t + 1 \)

Substituting these into the formula allows us to evaluate the integral and find the total length of the curve over the given interval.
The result is an exact measurement of the path's length between the starting and ending points.

Riemann Sums

Riemann sums are a key concept in calculus used to approximate the integral of a function over an interval. Think of them as summing up the areas of rectangles to estimate the area under a curve.
In this exercise, Riemann sums are used to approximate the length of the curve by breaking it into straight-line segments, making it easier to measure in parts. When you plot these line segments with increasing partition points (\(n=2,4,8\)), you're visually using Riemann sums to approach the real curve.
The Euclidean distance formula measures the length of each segment:\[\sqrt{(x(t_{i+1}) - x(t_i))^2 + (y(t_{i+1}) - y(t_i))^2}\]Summing these distances gives an approximation of the curve's length. As \(n\) increases, meaning more segments and smaller intervals, the approximation refines and gets closer to the actual length determined by the integral. This approach demonstrates how Riemann sums converge to the true integral value as partitions increase.