Problem 68 - Calculus Single Variable

Question

In a computer algebra system, a right endpoint approximation can be implemented by means of a one-line command. For example, if the real numbers \(a\) and \(b,\) the positive integer \(N,\) and the function \(f,\) have been defined, then the command $$ \operatorname{evalf}\left((b-a) / N^{*} \operatorname{add}(f(a+j *(b-a) / N), j=1 \ldots N)\right) $$ calculates formula \((5.1 .9)\) in Maple. In Exercises \(68-71,\) an interval \([a, b],\) and a function \(f\) are given. The function is positive on \((a, b)\) and it is 0 at the endpoints. Approximate the area under \(y=f(x)\) and over the interval \([a, b]\) by using the right endpoint approximation, starting with \(N=25 .\) Increment \(N\) by 25 until the first two decimal places of the sum remain the same for three consecutive calculations. Figure 12 shows a Maple implementation for the function \(f(x)=1-x^{x}, 0 \leq x \leq 1\). (This procedure does not guarantee two decimal places of accuracy. Section 5.7 presents several methods that can be used to achieve a prescribed accuracy.) $$ f(x)=\sqrt{16-x^{4}},-2 \leq x \leq 2 $$

Step-by-Step Solution

Answer

Calculate the area using increments of 25 for \(N\) until the result stabilizes.

Step 1: Define the Expression

Set up the expression for right endpoint approximation. For our function and interval, the Maple command is:\[ \operatorname{evalf}\left((b-a) / N^{*} \operatorname{add}(\sqrt{16-(a+j \cdot (b-a) / N)^4 }, j=1 \ldots N)\right) \] where \(a = -2, b = 2\). Start with \(N = 25\).

Step 2: Calculate Area for N = 25

Substitute \(N = 25\), \(a = -2\), and \(b = 2\) into the command: \[ \operatorname{evalf}\left((2-(-2)) / 25^{*} \operatorname{add}(\sqrt{16-((-2)+j \cdot (4/25))^4 }, j=1 \ldots 25)\right) \] Compute this to find the approximate area.

Step 3: Increase N to 50 and Calculate Again

Increment \(N\) to 50 and compute the area using the right endpoint approximation, as described previously. Substitute \(N = 50\) into the expression and calculate the result to an appropriate precision.

Step 4: Repeat for N = 75, 100, etc.

Continue increasing \(N\) by 25 each time, calculating the area for \(N = 75, 100, ...\). Evaluate the expression for each \(N\) using the right endpoint approximation method.

Step 5: Convergence Check

After each new calculation, check if the first two decimal places remain the same as the previous two calculations. As soon as this condition is met for three consecutive \(N\) values, conclude the process.

Key Concepts

Right Endpoint Approximation

When calculating the area under a curve, numerical integration techniques like the Right Endpoint Approximation are often used for simplicity, especially when dealing with non-polynomial functions. This method estimates the integral of a function over a given interval by dividing that interval into smaller segments and using the function's value at the right endpoint of each segment to approximate the area under the curve. For each subinterval, the height of the rectangle is determined by the function value at the right endpoint, and the width is the length of the subinterval.

• This method is simple and easy to implement.
• It may not always give an accurate result, especially when the function is rapidly changing.
• It is particularly useful as a first-step approximation method.

By increasing the number of intervals (i.e., increasing the value of \(N\)), the approximation becomes more accurate as the rectangles become narrower, better conforming to the curvature of the graph. But be aware, this method, while useful, does not always ensure precise results.

Convergence

Convergence in the context of numerical integration refers to the process by which a sequence of approximations approaches the true value of the integral as more intervals are used. For right endpoint approximation, convergence is witnessed when successive calculations produce values that become increasingly closer to each other. Practically, it is identified by observing when the calculated area value stabilizes to a consistent number of decimal places as \(N\) increases.
Here are a few aspects to understand about convergence:

• Stable results lead to greater confidence in the approximation's accuracy.
• Rapid convergence means fewer computations for a sufficiently accurate result.
• Some functions might require a significantly larger \(N\) for the values to stabilize.

For the exercise at hand, convergence is checked by calculating until the results keep the same first two decimal places over three consecutive \(N\) values. This ensures the approximation stops being overly sensitive to further increments in \(N\).

Maple Programming

Maple is a powerful tool for performing algebraic calculations symbolically and numerically. In this context, Maple can evaluate complex mathematical expressions, like the right endpoint approximation, in just a single line of command. This efficiency is particularly valuable when iterating calculations to achieve numerical convergence.
To use Maple for this exercise:

• Define the function and interval over which the area is calculated.
• Use Maple's syntax such as evalf and add to evaluate the area approximation.
• Iterate over increasing \(N\) values by updating the command and checking the output for stability.

For example, with input parameters like \(a = -2\), \(b = 2\), and increasing \(N\), the command simplifies complex iterations, minimizing errors and saving time. This ease of programming and accuracy makes Maple an ideal choice for engineering and mathematical computations.

Prescribed Accuracy

Prescribed accuracy involves determining how precise the numerical computation results should be and consistently achieving this level of precision. In numerical approximations, such as the right endpoint approximation, obtaining a prescribed accuracy requires careful attention to convergence criteria and the method used.
Here’s how you ensure prescribed accuracy:

• Identify the decimal precision needed (e.g., two decimal places).
• Use convergence checks to stabilize results to this precision.
• If the chosen method isn't converging effectively, explore alternative numerical methods like Simpson's Rule or the Trapezoidal Rule, which might provide faster or more reliable accuracy.

In the given exercise, patience and precision in monitoring decimal stability across calculations ensure the desired accuracy is met without overextending computational resources. By matching the level of approximation desired with actual results, students can assure the results' reliability, fitting prescribed standards for accuracy.