Problem 19 - Precalculus: Mathematical for Calculus

Question

Approximating Area with a Calculator When we approximate areas using rectangles as in Example \(1,\) then the more rectangles we use the more accurate the answer. The following TI- 83 program finds the approximate area under the graph of \(f\) on the interval \([a, b]\) using \(n\) rectangles. To use the program, first store the function \(f\) in \(Y_{1}\). The program prompts you to enter \(\mathrm{N}\), the number of rectangles, and \(A\) and \(B\), the endpoints of the interval. (a) Approximate the area under the graph of \(f(x)=x^{5}+2 x+3\) on \([1,3]\) using \(10,20\) and 100 rectangles. (b) Approximate the area under the graph of \(f\) on the given interval using 100 rectangles. (i) \(f(x)=\sin x, \quad\) on \([0, \pi]\) (ii) \(f(x)=e^{-x^{2}}, \quad\) on \([-1,1]\) \(P R O G R A M: A R E A\) : Prompt \(N\) : Prompt \(A\) : Prompt B \(:(B-A) / N \rightarrow D\) \(: 0 \rightarrow s\) \(: A \rightarrow x\) : For \((K, 1, N)\) \(: x+D \rightarrow x\) \(: \mathrm{S}+\mathrm{Y}_{1} \rightarrow \mathrm{S}\) : End \(: D \star S \rightarrow S\) : D is \(p\) "AREA IS" : D i sp \(\mathrm{S}\)

Step-by-Step Solution

Answer

Use more rectangles for better area approximation. Compare for different functions.

Step 1: Understanding the Taylor Approximation Using Rectangles

To approximate the area under a curve, divide the interval \( [a, b] \) into \( n \) equal subintervals, each of width \( D = \frac{(b-a)}{n} \). For each subinterval, the height of the rectangle is equal to the function value at the starting point of the subinterval.

Step 2: Setting Up the Program

Input the function into \( Y_1 \). Begin the program by entering the number of rectangles (\( N \)), and the interval endpoints (\( A \) and \( B \)). The calculator computes the step size \( D \) and sets the initial sum to \( 0 \).

Step 3: Running the Loop for Rectangle Heights

The loop iterates \( K \) from \( 1 \) to \( N \). During each iteration, increment \( x \) by \( D \) and add the value of \( Y_1 \) (the function at \( x \)) to the sum \( S \). This sum represents the sum of the heights of all rectangles.

Step 4: Computing the Approximate Area

Multiply the final sum \( S \) by \( D \) to get the approximated area under the curve, which is then displayed as the final result. \( S = D imes S \).

Step 5: Applying the Steps to the Given Function and Points

For \( f(x) = x^5 + 2x + 3 \) on \( [1, 3] \), run the program with \( N = 10 \), \( N = 20 \), and \( N = 100 \). Record the approximate area for each run. Repeat with \( f(x) = \sin x \) over \( [0, \pi] \) and \( f(x) = e^{-x^2} \) over \( [-1, 1] \) using \( N = 100 \).

Step 6: Interpretation of Results

The calculated approximate areas for increasing \( N \) indicate improved accuracy. Compare results across different \( N \) and functions to observe how approximation accuracy improves with more rectangles.

Key Concepts

Rectangles

Approximating the area under a curve using rectangles is a foundational technique in calculus and numerical analysis. This method is also known as the Riemann Sum. The basic idea is to divide the area under the curve into smaller sections using rectangles, which makes calculations more manageable.

For each rectangle:

• The width is determined by dividing the total interval [\(a, b\)] by the number of rectangles \(n\).
• The height is taken from the value of the function at a specific point within each interval.

By summing the areas of all rectangles, you get an approximation of the integral, or the area under the curve. More rectangles typically lead to a more accurate approximation as they represent the curve more closely.

In practice, as the number of rectangles increases, the approximation becomes closer to the true area. Therefore, selecting an appropriate number of rectangles is crucial for achieving a precise result.

TI-83 Program

The TI-83 calculator program is a powerful tool for performing these kinds of complex calculations. It helps simplify the process of approximating the area under a curve by automating repetitive calculations.

Here's how the TI-83 program works:

• First, you enter the function into the calculator's function list \(Y_1\).
• Next, the program prompts you to input the number of rectangles \(N\), as well as the interval boundaries \(A\) and \(B\).
• The program calculates the step size \(D\), which is the width of each rectangle, by dividing \((B-A)\) by \(N\).

Once set up, the program uses a loop to calculate the sum of the rectangle heights and multiplies this sum by their width to estimate the total area under the curve. This helps improve accuracy and saves you from the tedium of manual computation.

Function Approximation

Function approximation is the task of estimating a complicated function through simpler means. In the context of approximating areas, rectangles serve as the "building blocks" that simplify the function into manageable segments.

Here's the process:

• Begin by choosing a function, say \(f(x)\), which you want to approximate over an interval.
• The main focus is to approximate the integral of \(f(x)\), which represents the area under the curve.
• The choice of using rectangles is ideal due to their simplicity in terms of geometric calculation \(\text{(area = width × height)}\).

This approach makes complex calculations more approachable and tractable. The accuracy of the function approximation greatly depends on the number of rectangles. More rectangles mean a closer approximation, because smaller gaps and less overestimation/underestimation errors occur between the rectangle tops and the actual curve.

Interval

The interval is a critical component when approximating the area under a curve, as it defines the range over which the function is integrated. In other words, the interval \([a, b]\) tells us the start and end points for our approximation.

Consider these steps:

• Identify the function you want to integrate, say \(f(x)\).
• Determine the interval \([a, b]\) over which you need the approximation. This is your area of interest.
• The interval is divided into equal parts by the rectangles, and each part represents one rectangle.

The choice of interval influences the approximation because it affects the total number of rectangles (based on how the interval is divided) and the starting/ending points for calculating each rectangle's height. It is important to carefully define your interval to ensure that the approximation reflects the desired section of the curve.