Problem 71 - 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.) $$ f(x)=\sin ^{2}(x), 0 \leq x \leq \pi $$

Step-by-Step Solution

Answer

Calculate right endpoint sums with increasing \(N = 25, 50, \ldots\) until results in the first two decimal places stabilize over three consecutive results.

Step 1: Understand the Problem

We are given the function \(f(x) = \sin^2(x)\) over the interval \([0, \pi]\). We need to approximate the area under the curve using the right endpoint approximation, starting with \(N = 25\) and increasing by 25 until the first two decimal places of the result stabilize for three consecutive calculations.

Step 2: Set Up the Right Endpoint Approximation Formula

The formula for the right endpoint approximation is: \[\text{Area} \approx \frac{b-a}{N} \sum_{j=1}^{N} f\left(a + j \cdot \frac{b-a}{N}\right)\]Here, \(a = 0\), \(b = \pi\), and \(f(x) = \sin^2(x)\).

Step 3: Implement the Formula with N=25

First, calculate with \(N = 25\). The step size is \(\Delta x = \frac{\pi - 0}{25} = \frac{\pi}{25}\). Then, calculate the sum using right endpoints: \[\sum_{j=1}^{25} \sin^2\left(j \cdot \frac{\pi}{25}\right)\]

Step 4: Calculate the Approximation for N=25

Evaluate the sum and then multiply by \(\Delta x\) to approximate the area: \[\text{Area for } N=25 \approx \frac{\pi}{25} \sum_{j=1}^{25} \sin^2\left(j \cdot \frac{\pi}{25}\right)\]

Step 5: Repeat for N=50, N=75, etc.

Repeat the calculations for \(N = 50\), \(N = 75\), and so on:- Calculate \(\Delta x = \frac{\pi}{N}\) and the sum of \(\sin^2\left(j \cdot \frac{\pi}{N}\right)\) from \(j=1\) to \(N\).- Multiply by \(\Delta x\) for each \(N\).

Step 6: Check for Decimal Stability

Continue increasing \(N\) by 25 until the calculated area has the same first two decimal places for three consecutive calculations. This will involve evaluating sums and areas for several \(N\) values: \(N = 100, 125, \ldots\) until stable.

Key Concepts

Numerical Integration

Numerical integration is a method used when we want to find the area under a curve, especially when a function is complex or doesn't have an elementary antiderivative. It's essentially about estimating the integral of a function using discrete data points. For many real-world functions, especially those involving complex mathematics like trigonometric or non-linear functions, finding an exact integral might not always be feasible.
Instead, numerical integration offers practical approximations through different methods. In this exercise, we use the right endpoint approximation as a numerical integration technique. This involves dividing the interval into smaller sub-intervals and using the function's value at the right end of each subdivision to create rectangles whose areas can be summed up to approximate the area under the curve. While not exact, it is a powerful method, easy to implement, and improves in accuracy with smaller intervals and more data points.

• Approximations are essential in calculus, physics, and engineering.
• The right endpoint method is one of several methods, including the trapezoidal and Simpson's rules.

Interval Approximation

Interval approximation involves breaking the main interval \(a, b\) down into smaller parts to make calculations more manageable and precise. In the right endpoint approximation, the interval is divided into \(N\) smaller sections. Each of these sections has a width \(\Delta x = \frac{b-a}{N}\).
The accuracy of the approximation generally increases as you use more intervals (higher \(N\)). This means the function is being sampled at more points, smoothing out jagged edges or irregularities in the function's graph. In simpler terms, dividing the interval into more pieces gives a closer estimate of the true area under the curve.
In the specific example we worked on, we started with \(N=25\), and then gradually increased up to when the results stabilized, ensuring a more reliable approximation.

• Larger \(N\) values usually give better area estimates.
• Each subinterval adds detail to the final approximation.

Convergence Criteria

Convergence criteria determine when our numerical approximation is precise enough. This concept dictates the endpoint when subdividing the intervals no longer gives significantly different results. Convergence can be checked by observing the decimal stability of results after several iterations of increasing \(N\).
In simpler terms, in this exercise, we check that the first two decimal places of the approximated area value remain consistent for three sequential calculations with different \(N\) values. When this condition is met, we say that the sequence has converged and we've achieved a sufficiently accurate approximation.
This practical method ensures that we don't perform unnecessary calculations and waste computational resources. It helps achieve a balance between accuracy and efficiency.

• Convergence ensures we're getting valuable results without overwork.
• It tells us when to stop refining our numerical model.

Trigonometric Function Integration

Integrating trigonometric functions often requires special attention because these functions have periodic nature and unique properties. In this exercise, the function involved is \(f(x) = \sin^2(x)\), defined over the interval \[0, \pi\].
Integrating trigonometric functions, especially ones like \(\sin^2(x)\) which don't have a straightforward antiderivative, makes numerical methods indispensable. These methods bypass the need for finding a complex antiderivative and allow us to estimate the area under the curve using evaluations of the function at key points.
Trigonometric functions are used widely in physics and engineering, making understanding their integration crucial. By using numerical techniques here, students can grasp the practical side of calculus where direct integration is not feasible.

• Numerical integration of trigonometric functions helps handle complex periodic equations.
• These methods are applicable across various scientific fields.