Problem 58 - Thomas Calculus

Question

Use numerical integration to estimate the value of $$ \pi=4 \int_{0}^{1} \frac{1}{1+x^{2}} d x $$

Step-by-Step Solution

Answer

The estimated value of \(\pi\) is 3.1312.

Step 1: Identify the Task

We are asked to estimate the value of \(\pi\) using numerical integration for the integral \(4 \int_{0}^{1} \frac{1}{1+x^{2}} \, dx\).

Step 2: Set up the Integral

The integral represents the quarter-circle formula for \pi\. We need to approximate the value: \( I = 4 \int_{0}^{1} \frac{1}{1+x^{2}} \, dx \).

Step 3: Choose a Numerical Integration Method

Decide on a numerical integration method. For this exercise, we'll use the trapezoidal rule as a simple method to approximate definite integrals.

Step 4: Trapezoidal Rule Formula

The trapezoidal rule is given by: \(\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]\), where \(x_i\) are equally spaced points in the interval.

Step 5: Divide the Interval

Divide the interval \([0, 1]\) into \(n\) equal sub-intervals. Choose \(n=4\) for simplicity: \(x_0 = 0\), \(x_1 = 0.25\), \(x_2 = 0.5\), \(x_3 = 0.75\), \(x_4 = 1\).

Step 6: Calculate Function Values

Evaluate \(f(x) = \frac{1}{1+x^2}\) at each of the points: \[\begin{align*}f(x_0) & = \frac{1}{1+(0)^2} = 1, \f(x_1) & = \frac{1}{1+(0.25)^2} = \frac{16}{17}, \f(x_2) & = \frac{1}{1+(0.5)^2} = \frac{4}{5}, \f(x_3) & = \frac{1}{1+(0.75)^2} = \frac{16}{25}, \f(x_4) & = \frac{1}{1+(1)^2} = \frac{1}{2}.\end{align*}\]

Step 7: Apply Trapezoidal Rule

Plug the function values into the trapezoidal rule formula:\[\int_{0}^{1} \frac{1}{1+x^{2}} \, dx \approx \frac{0.25}{2} \left[1 + 2 \left(\frac{16}{17} + \frac{4}{5} + \frac{16}{25}\right) + \frac{1}{2} \right]\].

Step 8: Simplify and Calculate

Simplifying the expression results in:\[\begin{align*}\int_{0}^{1} \frac{1}{1+x^{2}} \, dx & \approx 0.125 \times \left[1 + 2 \left(\frac{16}{17} + \frac{20}{25} + \frac{16}{25}\right) + \frac{1}{2} \right], \& \approx 0.125 \times \left[1 + 2(0.9412 + 0.8 + 0.64) + 0.5 \right], \& \approx 0.125 \times \left[1 + 2 \times 2.3812 + 0.5 \right], \& \approx 0.125 \times 6.2624, \& \approx 0.7828.\end{align*}\]

Step 9: Multiply to Estimate Pi

Multiply the result by 4 to estimate \(\pi\): \[\pi \approx 4 \times 0.7828 = 3.1312\].

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a numerical method used to approximate the definite integral of a function. It is particularly useful when an exact integration is difficult or impossible to achieve analytically. The idea is to estimate the area under the curve by dividing it into small sections that resemble trapezoids, then summing up these areas to get an overall approximation. This method is grounded in the concept of calculus and can be a practical approach when dealing with many real-world applications.

The formula for the trapezoidal rule is derived from the concept of the area of a trapezoid: \[\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]\]

Here's how it works:

• Divide the interval \[a, b\] into \(n\) equal sub-intervals.
• Calculate the function values at each of these interval points.
• Apply the trapezoidal formula to approximate the integral.

This approximation gets better as the number of divisions, \(n\), increases. However, it's often a balancing act between computational efficiency and desired precision.

Definite Integrals

Definite integrals are a fundamental concept in calculus that represents the exact area under a curve from one point to another on a graph of a function. Specifically, a definite integral \(\int_{a}^{b} f(x) \, dx\) calculates the area under the curve \(f(x)\) between two bounds \(a\) and \(b\).

These integrals are crucial in various fields such as physics for computing work done by forces, in economics for calculating total profit over a time period, and many other areas. In essence, the definite integral gives you the total accumulation of the function's value as it moves from one point to another.

To compute definite integrals exactly, we need the antiderivative or an analytical form of the function. But, when this isn't accessible, numerical integration methods like the trapezoidal rule become valuable choices. They provide a way to estimate the integral's value with practical accuracy.

Approximation Methods

Approximation methods in calculus, such as the trapezoidal rule, are techniques used to find the approximate value of a definite integral when it's either difficult or impossible to integrate a function analytically. These methods are integral to applied mathematics and help solve real-world problems where precision isn't the highest priority or where analytic solutions are intractable.

There are several popular approximation methods:

• **Trapezoidal Rule:** Approximates the area under the curve using trapezoids.
• **Simpson's Rule:** Uses parabolas to approximate the areas, usually providing more accuracy.
• **Midpoint Rule:** Approximates the integral using rectangles whose heights are the values of \(f(x)\) at midpoints.

Each method has its own strengths and weaknesses, often trading complexity for accuracy or computational simplicity. Choosing the right method involves considering the specific function's behavior and the desired accuracy of the approximation. For many applications, especially when time or resources are limited, these methods give us good enough answers quickly and efficiently.