Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Parabolic Rule with \(n=8\) to approximate the area of the region trapped between \(y=1-e^{-x^{2}}\) and \(y=e^{-x^{2}} .\)

Step-by-Step Solution

Answer

Use Simpson's Rule with defined bounds, evaluate at points, then apply the rule.

Step 1: Understand the Problem

We are tasked with finding the area between two curves, specifically between the functions \( y = 1 - e^{-x^2} \) and \( y = e^{-x^2} \). The parabolic rule, also known as Simpson's Rule, is to be used with \( n=8 \).

Step 2: Establish Limits of Integration

First, we need to determine the interval over which to integrate. The problem does not specify the bounds of integration, so we typically assume it to be from the point where the functions intersect. Both functions intersect where their equations are equal: \( 1 - e^{-x^2} = e^{-x^2} \), which simplifies to \( x=0 \). As no further bounds are provided, we can set arbitrarily an interval such as \(-a\) to \(a\).

Step 3: Set Up Simpson's Rule

Simpson's Rule is given by the formula: \[ S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n) \right] \] where \(\Delta x = \frac{b-a}{n}\), \(a\) and \(b\) are the bounds, and \( n\) is the number of subintervals. Here \( a \) and \( b \) need defining based on the chosen range.

Step 4: Calculate Step Size \( \Delta x \)

With our boundaries set from \( -a \) to \( a \) and \( n=8\), \( \Delta x = \frac{b-a}{n} = \frac{c - (-c)}{8} = \frac{2c}{8} = \frac{c}{4} \).

Step 5: Evaluate Functions at Specific Points

Evaluate the function to be integrated, \( f(x) = (1 - e^{-x^2}) - e^{-x^2} = 1 - 2e^{-x^2} \), at each sample point. Sample points are determined by the range of integration and are spaced \( \Delta x \) apart. Calculate \( f(x_i) \) where \( x_i = -a + i \Delta x \) for each \( i \) from 0 to 8.

Step 6: Compute Approximation Using Simpson's Rule

Use the values obtained in the previous step to substitute into the Simpson's Rule formula and compute the integral. Ensure the calculations take into account the coefficients in Simpson's Rule: 1 for endpoints, 4 for odd-indexed points, and 2 for even-indexed points.

Key Concepts

Area Between Curves

Understanding the area between curves is a fundamental topic in calculus. When we talk about the area between two curves, we're interested in finding the total area that lies between the graphs of two functions over a specified interval. This often involves finding the difference between the upper and lower functions.For instance, if you have two functions, say, \( y = f(x) \) and \( y = g(x) \), you'll compute the area between these curves by integrating the difference \( f(x) - g(x) \) over a specific interval. Breaking it down:

• Identify where the two functions intersect, as these points determine the integral's limits.
• Set up the integral of the difference of the upper function and lower function within these limits.

This concept is crucial when using Simpson's Rule to approximate areas, as seen in problems where exact intersection points can guide your integration intervals.

Numerical Integration

Numerical integration is a technique used to approximate the value of an integral. This is particularly useful when the integral cannot be solved analytically due to complex functions or undefined boundaries. By employing numerical methods, we can obtain approximate solutions that are both practical and efficient. In the context of Simpson's Rule, numerical integration involves:

• Dividing the total interval into smaller subintervals.
• Using a polynomial (in this case, a quadratic or parabolic) to approximate the area under the curve for each subinterval.

The beauty of numerical integration lies in its adaptability to complex shapes and varying functions. It bridges the gap between purely theoretical calculus and practical computation, making it essential for solving real-world calculus problems.

Parabolic Rule

The Parabolic Rule, more commonly known as Simpson's Rule, is a method in numerical integration that remarkably approximates the area under a curve by using parabolic arcs. This rule is particularly powerful because it provides a weighted average of the midpoint (the value of the function at the middle of the interval) and the endpoint values, yielding a very accurate approximation.Key steps in Simpson's Rule:

• Divide the interval into an even number of subintervals (since \( n \) must be even).
• Calculate the width of each subinterval, \( \Delta x = \frac{b-a}{n} \).
• Apply the Simpson’s Rule formula: \[ S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n) \right] \]

This weighted sum accounts for the curve's shape, allowing a parabolic approximation that closely follows the actual area under the curve, thus embodying the efficiency of numerical methods.

Calculus Problems

Solving calculus problems often requires a deep understanding of a variety of concepts, from finding areas and solving differential equations to dealing with limits and continuity. Each problem is unique and can involve different techniques or formulas. Specific to our discussion:

• Recognize which method is best suited for the problem at hand—analytical or numerical.
• Understand that even with a strong grasp of algebraic and geometric concepts, calculus problems demand logical reasoning and sometimes require trial and error with various methods to find the best solution.

In numerical methods like Simpson's Rule, calculus problems become manageable, allowing for approximations where exact solutions are elusive. Practice and familiarity with these tools transform complex calculus scenarios into solvable exercises.