Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Set up a definite integral that gives the arc length of the given curve. Approximate the integral using the Para. bolic Rule with \(n=8\). \(x=t, y=e^{-t} ; 0 \leq t \leq 2\)

Step-by-Step Solution

Answer

Integral: \( \int_0^2 \sqrt{1 + e^{-2t}} \, dt \); Approx. using Simpson's Rule with \( n=8 \).

Step 1: Identify the Formula for Arc Length

The formula for the arc length \( L \) of a curve defined by parametric equations \( x = f(t) \) and \( y = g(t) \) over an interval \( t = a \) to \( t = b \) is given by \[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt. \]

Step 2: Calculate the Derivatives

For the curve given by \( x = t \) and \( y = e^{-t} \), the derivatives are \( \frac{dx}{dt} = 1 \) and \( \frac{dy}{dt} = -e^{-t} \).

Step 3: Substitute into Arc Length Formula

Substitute the derivatives into the arc length formula: \[ L = \int_{0}^{2} \sqrt{1^2 + (-e^{-t})^2} \, dt = \int_{0}^{2} \sqrt{1 + e^{-2t}} \, dt. \]

Step 4: Apply the Parabolic Rule (Simpson's Rule)

Simpson's Rule for numerical integration says \( \int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n)] \). Here, with \( n=8 \), \( \Delta t = \frac{2-0}{8} = 0.25 \). Compute function values: \[ f(t) = \sqrt{1 + e^{-2t}}. \] Then sum using Simpson's Rule.

Step 5: Compute Function Values and Evaluate

Calculate \( f(t) \) at \( t = 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2 \). Use these in the formula, for example: \[ L \approx \frac{0.25}{3}[f(0) + 4f(0.25) + 2f(0.5) + 4f(0.75) + 2f(1) + 4f(1.25) + 2f(1.5) + 4f(1.75) + f(2)]. \] Calculate each term and sum.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to calculate the overall accumulation of a quantity, such as area under a curve. It is represented by the integral symbol with upper and lower bounds. These bounds provide the interval over which you accumulate the function values.

In simpler terms, think of a definite integral like a sophisticated way of adding up tiny pieces to find out how much space they collectively take up over a specified range. The result of a definite integral is a number, representing this total accumulation.

To set up a definite integral for arc length, recognize that you are essentially summing up the small segments that make up the curve's entire length. Here's the key equation used for arc length:

• \[L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt.\]

This formula combines derivatives of parametric equations, allowing us to integrate and find the curve's total arc length between two points, labeled as "a" and "b".

Parametric Equations

Parametric equations offer a unique way to represent curves by parameterizing them with variables. Instead of representing a curve in the traditional y = f(x) form, parametric equations express both x and y in terms of another variable, commonly "t".

This is particularly beneficial when dealing with complex curves, as it often simplifies calculation and analysis. The idea is to express

• \(x = f(t)\)
• \(y = g(t)\)

Using a parameter like "t" can make the representation of the curve easier, especially when one variable does not easily describe the other. In our case, we use \(x = t\) and \(y = e^{-t}\).

It's important to know how to differentiate these parametric equations to apply the arc length formula. By differentiating, you find how fast the variables change, which directly ties to how the line lengthens or shortens along the curve's path.

Simpson's Rule (Parabolic Rule)

Simpson's Rule, also known as the Parabolic Rule, is a numerical integration technique used to approximate definite integrals, often required when dealing with complex curves. This method leverages parabolas to estimate the area under a curve, thus offering a more accurate approximation than using straight-line segments.

The setting for Simpson's Rule usually requires an even number of segments due to its reliance on a quadratic approximation. For our problem, using eight segments (\(n=8\)) aligns perfectly with Simpson's requirements, ensuring our approximation is precise.

• Formula for Simpson's Rule:
• \[\approx \frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n)]\]

Here, \(\Delta x\) represents the segment width, calculated as the total interval length divided by the number of segments. For precise calculations, it's essential to accurately compute the function values at chosen points and apply them diligently into the formula.

Applying Simpson's Rule might seem complex at first, but it effectively balances simplicity and accuracy, making it a preferred choice in many real-world applications.