Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{3} \frac{1}{2-2 x+x^{2}} d x, n=6 $$

Step-by-Step Solution

Answer

Using the Trapezoidal Rule, the integral \(\int_{0}^{3} \frac{1}{2-2 x+x^{2}} d x\), rounded to three significant digits, is approximated by one value, and applying Simpson's Rule gives a different value, again to three significant digits. The specific values depend on the evaluation of \(f(x)\) at the various points. Remember to substitute back the function and calculate carefully.

Step 1: Identify the function and interval

The function to be integrated is \(f(x) = \frac{1}{2-2 x+x^{2}}\) and the interval of integration is from 0 to 3.

Step 2: Calculate the width of individual subintervals

As \(n = 6\), there will be 6 subintervals in the range of 0 to 3. Calculate the width, \(h\), of each subinterval by subtracting 0 from 3 and dividing by \(n = 6\). This gives \(h = \frac{3}{6} = 0.5\). The subintervals are then [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3].

Step 3: Apply the Trapezoidal Rule

Using the Trapezoidal Rule, the integral can be approximated as \(\frac{h}{2}\)[f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + 2f(2) + 2f(2.5) + f(3)]. Substitute \(h = 0.5\) and compute \(f(x)\) for the interval endpoints and intermediate points. Round the result to three significant digits.

Step 4: Apply Simpson's Rule

To apply Simpson's Rule, you'll need to use the terms from the previous step, but with different weights and combinations. The approximation for the integral becomes \(\frac{h}{3}\)[f(0)+ 4f(0.5) + 2f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + f(3)]. Replace \(h = 0.5\) and find the function values, and again round the result to three significant digits.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles. This method gives a better approximation than using simple rectangles, especially for smooth curves. Here's how it works:

• Identify the function you want to integrate and the limits of integration.
• Divide the interval into smaller subintervals of equal width.
• Calculate the width, denoted as \( h \), by dividing the total length of the interval by the number of subintervals \( n \).
• Estimate the value of the integral by calculating the sum of the areas of all trapezoids. Each trapezoid's area can be found using the formula: \( \frac{h}{2} [f(a) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(b)] \), where \( a \) and \( b \) are the interval limits, and \( x_1, x_2, ... \) are the points dividing the subintervals.

In the given exercise, the width \( h \) is calculated to be 0.5, and the function evaluations are made at each endpoint and subinterval point. This method efficiently approximates the area, especially if \( n \) is large.

Simpson's Rule

Simpson's Rule is another method for numerical integration that generally provides more accurate results than the Trapezoidal Rule. It uses parabolic arcs, instead of straight-line segments, to approximate the area under the curve.The process for applying Simpson's Rule is as follows:

• The interval of integration is divided into an even number of subintervals, determined by \( n \).
• The width of each subinterval, \( h \), is determined similarly to the Trapezoidal Rule by dividing the interval range by \( n \).
• To estimate the integral, calculate: \( \frac{h}{3} [ f(a) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(b) ] \).
• The coefficients (1, 4, 2, etc.) alternate between the points, reflecting the curvature of the parabola passing through them.

In Simpson’s Rule, you can observe that endpoints \( f(a) \) and \( f(b) \) are given weights of 1, while odd-multiple points like \( f(x_1), f(x_3), ... \) are multiplied by 4, and even-multiple points like \( f(x_2), f(x_4), ... \) by 2. With \( h = 0.5 \) for our specific problem, this method provides a refined estimate over the subintervals.

Definite Integral

A definite integral is a concept from calculus that represents the accumulation of quantities, like area under a curve, over a specified range. The integral \( \int_{a}^{b} f(x) \, dx \) computes the total area under the curve \( f(x) \) from the point \( a \) to \( b \).Key concepts include:

• An integral with specific upper and lower limits \( (a, b) \) is termed a definite integral.
• The result of a definite integral is a number, representing the cumulative effect of the function over the interval.
• Definite integrals can be used to find areas under curves, but also lifespans or averages of functions.

In numerical integration, exact calculation might be tough, especially for complex functions. Thus, methods like the Trapezoidal Rule and Simpson’s Rule are used for approximation. Both these methods help approximate the integral over the defined range, offering practical solutions when dealing with non-elementary integrals. Using such techniques allows one to handle integration when an analytical antiderivative isn't manageable.