Problem 26 - Calculus Early Transcendentals

Question

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.5 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{0}^{8} e^{-2 x} d x=\frac{1-e^{-16}}{2}\)

Step-by-Step Solution

Answer

Question: Using the Midpoint and Trapezoid Rules, approximate the definite integral \(\int_{0}^{8} e^{-2 x} d x\) for \(n = 4, 8, 16,\) and \(32\). Compare the approximations and errors of the two methods. Solution: Set up a table for calculating the Midpoint Rule (M) and Trapezoid Rule (T) errors for different values of n, along with their errors with respect to the exact integral value, \(\frac{1-e^{-16}}{2}\). Analyze the results and notice how the rules' approximations improve with increasing n.

Step 1: Determine the function and interval

The function we are working with is \(f(x) = e^{-2x}\), and the interval of integration is \([0, 8]\). The exact integral value is \(\frac{1 - e^{-16}}{2}\).

Step 2: Write the formulas for Midpoint and Trapezoid Rules

The formulas for the Midpoint and Trapezoid Rules are as follows: Midpoint Rule: \(M_n = \Delta x \sum_{i=1}^n f(x_i^*)\) Trapezoid Rule: \(T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]\) where \(\Delta x = \frac{b - a}{n}\), and the interval of integration is \([a, b]\).

Step 3: Set up a table to calculate approximations and errors

Create a table with columns for \(n\), \(\Delta x\), \(M_n\), \(|M_n - Exact|\), \(T_n\), and \(|T_n - Exact|\).

Step 4: Calculate Midpoint and Trapezoid Rule approximations for different n values

For each value of \(n = 4, 8, 16, 32\), do the following: 1. Calculate the \(\Delta x = \frac{8 - 0}{n}\). 2. Compute \(M_n\) using the Midpoint Rule formula. 3. Calculate the error \(|M_n - Exact|\). 4. Compute \(T_n\) using the Trapezoid Rule formula. 5. Calculate the error \(|T_n - Exact|\). Fill in the table with the calculated values.

Step 5: Interpret the results

Compare the Midpoint and Trapezoid Rule approximations to the exact integral value for each value of n. Analyze the errors and notice how the rules' approximations improve with increasing n.

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical method to approximate the value of an integral. It works by dividing the interval into small, equal segments and calculating an average value by using the midpoint of each segment.

• The formula used is: \[ M_n = \Delta x \sum_{i=1}^n f(x_i^*) \]
• This approach works by summing the function values at the midpoints of each sub-interval, represented as \(x_i^*\).
• Here, \(\Delta x\) is the width of each sub-interval, calculated as \(\frac{b-a}{n}\), where \([a,b]\) is the interval length and \(n\) is the number of segments.

This rule is often chosen for its simplicity and moderate accuracy. By increasing \(n\), the approximation improves, making it useful for getting quick estimates, especially when an analytical solution is complex.

Trapezoid Rule

Another effective way to approximate integrals is the Trapezoid Rule. This method uses trapezoids instead of rectangles, providing a closer approximation by connecting the function's values at interval endpoints.

• The formula is given by: \[ T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + \ldots + 2f(x_{n-1}) + f(x_n)] \]
• The \(\Delta x\), similar to the Midpoint Rule, is the width of each interval.
• The multiplication by 2 for intervening terms accounts for the shared sides of adjacent trapezoids.

This method generally provides more accurate results compared to the Midpoint Rule for the same \(n\), particularly because it considers the area under the curve more comprehensively. As \(n\) increases, the approximation nears the exact value, indicating improved precision.

Error Analysis

Error analysis is crucial when applying numerical methods like the Midpoint and Trapezoid Rules. It helps quantify how close the approximation is to the true integral value.

• The error for the Midpoint Rule decreases approximately according to the behavior of \(O\left(\frac{1}{n^2}\right)\), meaning as \(n\) increases, the error reduces significantly.
• The Trapezoid Rule exhibits a similar error pattern with \(O\left(\frac{1}{n^2}\right)\), enhancing its accuracy with more subdivisions.

By calculating \(|M_n - \text{Exact}|\) and \(|T_n - \text{Exact}|\), you can track how close these methods are to the actual integral. Reducing error helps confirm the reliability of numerical calculations, rendering them valuable tools in estimating integrals where exact solutions are challenging.