Question

Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of \(n .\) a. Find the Trapezoid Rule approximations to the integral using \(n\) and \(2 n\) subintervals. b. Find the Simpson's Rule approximation to the integral using \(2 n\) subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(8 .\) c. Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with \(2 n\) subintervals. $$\int_{0}^{1} e^{2 x} d x, n=25$$

Short answer

Question: Compute the Trapezoid Rule approximation and Simpson's Rule approximation for the integral \(\int_{0}^{1} e^{2 x} d x\) with \(n = 25\) subintervals. Calculate the absolute errors for both approximations. Answer: To compute the approximations and absolute errors, follow these steps: 1. Determine the Trapezoid Rule approximations with \(n\) (25) and \(2n\) (50) subintervals, labeled as \(T_{25}\) and \(T_{50}\). 2. Use these Trapezoid Rule approximations to compute the Simpson's Rule approximation with \(2n\) subintervals, labeled as \(S_{50}\). 3. Calculate the exact value of the integral, which is \(\frac{1}{2}(e^2 - 1)\). 4. Compute the absolute errors for the Trapezoid Rule (\(|T_{50} - (\frac{1}{2}(e^2 - 1))|\)) and Simpson's Rule (\(|S_{50} - (\frac{1}{2}(e^2 - 1))|\)) approximations.

Step-by-step solution

Trapezoid Rule Approximations with \(n\) and \(2n\) subintervals

Given \(\int_{0}^{1} e^{2 x} d x\). Let's first compute the Trapezoid Rule approximation. We have, \(n = 25\). So for \(n\) subintervals, the width of each interval, \(\Delta x = \frac{1 - 0}{25} = \frac{1}{25}\). For \(2n\) subintervals, the width of each interval, \(\Delta x' = \frac{1 - 0}{50} = \frac{1}{50}\). Now, let's compute the Trapezoid Rule approximation using these widths: Trapezoid Rule = \(\frac{1}{2}\Delta x \left[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)\right]\). For \(n = 25\), let's call this approximation \(T_{25}\). For \(2n = 50\), let's call this approximation \(T_{50}\).

Simpson's Rule Approximation with \(2n\) subintervals

We know that the Simpson's Rule approximation can be obtained from the Trapezoid Rule approximations: Simpson's Rule = \(\frac{2}{3}T_{50} - \frac{1}{3}T_{25}\). Let's call this approximation \(S_{50}\).

Calculate Absolute Errors

First, we need to know the exact value of the integral. Compute the antiderivative of \(e^{2x}\), which is \(\frac{1}{2}e^{2x}\). Now, evaluate the antiderivative at \(1\) and \(0\) to find the exact value: \(\frac{1}{2}e^{2(1)} - \frac{1}{2}e^{2(0)} = \frac{1}{2}(e^2 - 1)\). Now, compute the absolute errors for the Trapezoid Rule and Simpson's Rule with \(2n\) subintervals: Absolute error for Trapezoid Rule: \(|T_{50} - (\frac{1}{2}(e^2 - 1))|\). Absolute error for Simpson's Rule: \(|S_{50} - (\frac{1}{2}(e^2 - 1))|\). By following these steps, we can compute the approximate values for the integral using the Trapezoid Rule and Simpson's Rule, as well as the absolute errors associated with these approximations.

Key concepts

Trapezoid Rule

The Trapezoid Rule is a numerical integration method used to approximate the value of a definite integral. It is particularly straightforward and useful when dealing with functions that are difficult or impossible to integrate analytically. The basic idea is to divide the area under the curve into a series of trapezoids rather than squares or rectangles.

The formula for the Trapezoid Rule is expressed as follows: \begin{align*}T_n &= \frac{1}{2}\Delta x \left[f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)\right],\end{align*}where \( n \) is the number of subintervals, \( \Delta x \) is the width of each subinterval, and \( f(x_i) \) are the function values at the endpoints and midpoints of the subintervals. The endpoints contribute only once to the sum, while all other points contribute twice. This method takes its name because each pair of points defines a trapezoid in the xy-plane.

For the given exercise, applying the Trapezoid Rule with \( n = 25 \) and then with \( 2n = 50 \) subintervals provides two approximations, \( T_{25} \) and \( T_{50} \), respectively. By using more subintervals, we can often achieve a better approximation.

Simpson's Rule

Simpson's Rule is another technique for approximate integration that typically provides more accurate results than the Trapezoid Rule for a given number of subintervals. It is based on the concept of replacing the function with a series of parabolas that pass through the function points, and then calculating the area under these parabolas.

The formula for Simpson's Rule, derived from the Trapezoid Rule, is given by:\[ S_{2n} = \frac{2}{3}T_{50} - \frac{1}{3}T_{25} \],where \( T_{50} \) is the Trapezoid Rule approximation with \( 50 \) subintervals, and \( T_{25} \) is the Trapezoid Rule approximation with \( 25 \) subintervals. This equation reflects a correction applied to the average of two Trapezoid Rule approximations to yield the more precise Simpson's Rule approximation, denoted as \( S_{50} \) in the given problem.

When using Simpson's Rule, it is important to note that the number of subintervals must be even because the rule integrates over pairs of intervals. In practice, Simpson's Rule tends to be significantly more accurate than the Trapezoid Rule if the function being integrated is well-behaved within the intervals.

Approximate Integration

Approximate integration is the process of estimating the value of a definite integral when an exact analytical solution is difficult or impossible to find. In applied mathematics and engineering, many integrals cannot be solved with elementary functions, making numerical methods invaluable.

The methods like the Trapezoid Rule and Simpson's Rule come into play as key tools for approximation. These methods work by simplifying the function into a shape that can be easily calculated—trapezoids for the Trapezoid Rule and parabolic arcs for Simpson's Rule.

While some functions can be integrated exactly, for most real-world scenarios, these numerical methods provide results that are sufficiently accurate for practical purposes. The balance between computational effort and accuracy is a major consideration when choosing the appropriate method for a given problem. As the number of subintervals increases, so does the accuracy of the approximation, but at the cost of increased computation time and resources.

Absolute Error in Numerical Integration

The absolute error in numerical integration is a measure of how much an approximate result differs from the true value of the integral. When we approximate an integral using techniques such as the Trapezoid Rule or Simpson's Rule, the absolute error quantifies the accuracy of these approximations.

For a given approximation \( A \) and the true value \( V \), the absolute error \( E \) is calculated as:\[ E = |A - V| \].

In the context of the given exercise, calculating the absolute errors for the Trapezoid Rule approximation \( T_{50} \) and Simpson's Rule approximation \( S_{50} \) involve subtracting these from the exact integral value and taking the absolute value of the result. As students use these methods, understanding and computing the absolute error is crucial because it gives an indication of reliability and guides them on whether they need to refine their calculations with additional subintervals or another method to achieve a desired level of accuracy.