Problem 37 - Thomas Calculus

Question

In Exercises \(31-36,\) 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 \(6.24 . )\) 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 appoximations for \(n=2,4,8\) with the actual compare= given by the integral. How does the actual length compare with the approximations as \(n\) increases? Explain your answer. $$ x=\frac{1}{3} t^{3}, \quad y=\frac{1}{2} t^{2}, \quad 0 \leq t \leq 1 $$

Step-by-Step Solution

Answer

The actual curve length is approximated more accurately as \(n\) increases, converging to the integral's result.

Step 1: Set up the parametric equations

Identify the given parametric equations for the curve: \[ x(t) = \frac{1}{3} t^3, \quad y(t) = \frac{1}{2} t^2 \]The range for the parameter \(t\) is from \(0\) to \(1\).

Step 2: Plot the curve and polygonal path

To plot the curve, use a graphing software or CAS to graph \(x(t)\) and \(y(t)\) within the interval \(0 \leq t \leq 1\). Then, superimpose the polygonal approximations for \(n=2, 4, 8\). Each \(n\) value represents partition points, which will create line segments connecting these points along the curve.

Step 3: Find lengths of line segments for n=2, 4, and 8

For each value of \(n\), calculate the length of each line segment using the Euclidean distance formula between points on the curve.For any partition point \(t_i\) and \(t_{i+1}\), the line segment length is given by:\[L_i = \sqrt{(x(t_{i+1}) - x(t_i))^2 + (y(t_{i+1}) - y(t_i))^2}\]Sum the lengths for each \(n\) to approximate the total curve length.

Step 4: Calculate curve length using an integral

The exact length of the curve from \(t=0\) to \(t=1\) is found using the arc length formula:\[L = \int_{0}^{1} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]Differentiate \(x(t)\) and \(y(t)\) with respect to \(t\):\[ \frac{dx}{dt} = t^2, \quad \frac{dy}{dt} = t \]Substitute these into the integral:\[L = \int_{0}^{1} \sqrt{(t^2)^2 + (t)^2} \, dt = \int_{0}^{1} \sqrt{t^4 + t^2} \, dt\]Evaluate the integral to find the exact length.

Step 5: Compare approximations with exact length

Calculate the integral from Step 4 to find the actual length. Compare this result with the summed lengths from Step 3 for \(n = 2, 4, 8\). As \(n\) increases, the approximation becomes closer to the exact length.

Key Concepts

Arc Length Computation

Computing the arc length of a parametric curve involves integrating to find the total distance along the curve. In our exercise, you determine the length by considering the parameters for

• x(t) = \( \frac{1}{3} t^3 \)
• y(t) = \( \frac{1}{2} t^2 \)
.

The range for the parameter \(t\) is between 0 and 1.To find the arc length, first differentiate each function with respect to \(t\):

• \( \frac{dx}{dt} = t^2 \)
• \( \frac{dy}{dt} = t \)

Next, plug these into the arc length formula:\[L = \int_{0}^{1} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt = \int_{0}^{1} \sqrt{t^4 + t^2} \, dt.\]This integral gives the exact length of the curve between \(t=0\) and \(t=1\). By solving this integral thoroughly, you obtain the precise measurement of the curve's path.

Polygonal Path Approximation

A polygonal path approximation is a technique used to estimate the arc length of a curve by breaking it into straight line segments. In the given exercise, this is done by plotting points along the curve for different \(n\) values (\(n=2, 4, 8\)). For each \(n\):

• Divide the interval \([0,1]\) into \(n\) equal subintervals.
• Evaluate the curve at endpoints of each subinterval to create vertices of the polygonal path.

To calculate the length of these segments, you use the Euclidean distance formula, \(L_i = \sqrt{(x(t_{i+1}) - x(t_i))^2 + (y(t_{i+1}) - y(t_i))^2}\). Summing these segment lengths for each value of \(n\) provides an approximation of the total curve length.
As \(n\) increases, the approximation becomes more accurate, closely resembling the curve's true length.

Integration Techniques

Integration plays a crucial role when computing the exact arc length of parametric curves. The integral \[L = \int_{0}^{1} \sqrt{t^4 + t^2} \, dt\]can be approached using a substitution, leading to more manageable algebraic manipulation. To perform this integration effectively:

• Look for substitutions that simplify the expression inside the square root to a basic form.
• Use integration by parts or partial fractions if the expression permits.

Advanced techniques may also involve transforming the integrand into a simpler, known form whose integral is already established. The mastery of these techniques allows you to derive exact lengths and solve complex equations, which are fundamental skills in calculus and beyond.