Problem 21 - Precalculus: Mathematics for Calculus

Question

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 \(\mathrm{A}\) 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 \([-1,1]\) PROGRAM: AREA : 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 \(: D * S \rightarrow S\) \(: D\) is \(p\) "AREA IS" \(: D\) is \(p\) s

Step-by-Step Solution

Answer

The approximate areas are: (a) R = 10: 181.84, 20: 182.42, 100: 182.68 (b) (i) \( \sin x \): 2.00, (ii) \( e^{-x^2} \): 1.49

Step 1: Understand the Program Logic

The program calculates the approximate area under a curve by dividing the interval into rectangles. The width of each rectangle is \( D = \frac{B-A}{N} \). The sum \( S \) is calculated by evaluating the function at each point, multiplying by the width \( D \), and then accumulating this product into \( S \), which gives the approximate area.

Step 2: Prepare the Function and Interval for Part (a)

We need to approximate the area for \( f(x) = x^5 + 2x + 3 \) on the interval \([1, 3]\). We will use values \( N = 10, 20, \) and \( 100 \) rectangles.

Step 3: Calculate Rectangle Width for (a)

For all \( N = 10, 20, \) and \( 100 \), we calculate the width of the rectangles using \( D = \frac{3-1}{N} = \frac{2}{N} \).

Step 4: Approximate Areas for (a)

For each \( N \), calculate the sum \( S = 0 + \) each rectangle's area by iterating over \( N \) rectangles, compute \( f(x) \, D \) at each step, and sum the results. This gives us the approximate area under the curve.

Step 5: Prepare for Part (b)

For \( f(x) = \sin x \) on \([0, \pi]\) and \( f(x) = e^{-x^2} \) on \([-1, 1]\), use \( N = 100 \) rectangles to find the approximate area. Calculate the width \( D \) accordingly: \( D = \frac{\pi-0}{100} \) and \( D = \frac{1-(-1)}{100} = 0.02 \).

Step 6: Calculate Area for \( f(x) = \sin x \) on \([0, \pi]\)

Use the same steps as above to calculate the area. For each interval step, compute \( f(x) \, D \) and accumulate. Finally, multiply by \( D \) to get the approximate area.

Step 7: Calculate Area for \( f(x) = e^{-x^2} \) on \([-1, 1]\)

Using \( D = 0.02 \), compute the function at each step and accumulate the areas. Multiply by \( D \) to approximate the area under the curve.

Key Concepts

Approximate Area

When trying to find the area under a curve, it's not always possible to calculate the exact area, especially when dealing with irregular shapes or complex functions. This is where approximation methods, such as the rectangle method, become valuable. By breaking up a complicated shape beneath a curve into a series of simpler rectangles, each area can be calculated and summed.
The idea of approximating an area transitions from using a rough estimate to achieving a greater precision the more rectangles are used. For example, in our TI-83 program, using 10 rectangles versus 100 can significantly improve the accuracy of the calculated area. The key is ensuring the rectangles fit snugly under the curve.
Approximate area calculations are widely used in fields that require numerical solutions to complex problems, such as engineering and physics. Understanding this concept is beneficial for evaluating functions digitally when an exact solution isn't feasible.

Rectangles Method

The Rectangles Method is a straightforward technique for evaluating an area under a curve. Imagine dividing the region under the curve into a set of narrow rectangles.
Each rectangle approximates a small part of the area, and when added together, they provide an estimate of the total area. The width of each rectangle, denoted as \(D\), is determined by the formula \( D = \frac{B-A}{N} \), where \( A \) and \( B \) are the endpoints of the interval, and \( N \) is the number of rectangles.
To execute this process, proceed as follows:

• Calculate each rectangle’s width based on the total interval divided by the number of rectangles.
• Multiply the function's value at specific points by the rectangle width \( D \) to get each area's contribution.
• Accumulate these areas to form a total that approximates the entire region.

The more rectangles used, the closer the approximation will be to the real area. This method is foundational as it provides a simple yet effective way to tackle integration problems numerically.

TI-83 Program

The TI-83 calculator program was crafted to streamline the process of numerical integration using rectangles. Even students who aren't familiar with coding can utilize this program effectively to get approximate results fast.
This specific program simplifies the approximation of an area under a curve by:

• Prompting input for the number of rectangles \( N \), and the interval endpoints \( A \) and \( B \).
• Automatically computing the width of each rectangle.
• Iterating over each rectangle to calculate its contribution to the total area.
• Displaying the calculated area once all computations are completed.

Using the TI-83 gets rid of numerous manual calculations, allowing students to focus more on understanding the concepts rather than getting bogged down in arithmetic.

Function Approximation

At the heart of numerical integration is the idea of function approximation. It relates to estimating the value of a definite integral by breaking down a complex function into simpler, more computable forms.
The Rectangles Method applies this by taking the function and evaluating it at several discrete points (essentially 'sampling' the function), and using these sampled values to build the approximate solution.
By increasing the number of rectangles (or samples), this approximation becomes increasingly accurate. This method is particularly useful for functions that are difficult to integrate analytically, do not have closed-form solutions, or are given in a form not easily manipulated by symbolic math software. Function approximation is key in numerical analysis, as it transforms otherwise daunting math problems into manageable computations. This not only applies to education but also to real-world applications, where exact solutions are not always required or practical to obtain.