Problem 36 - Applied Calculus

Question

(a) Graph \(f(x)=e^{-x^{2}}\) and shade the area represented by the improper integral \(\int_{-\infty}^{\infty} e^{-x^{2}} d x\) (b) Find \(\int_{-a}^{a} e^{-x^{2}} d x\) for \(a=1, a=2, a=3, a=5\). (c) The improper integral \(\int_{-\infty}^{\infty} e^{-x^{2}} d x\) converges to a finite value. Use your answers from part (b) to estimate that value.

Step-by-Step Solution

Answer

The improper integral converges to approximately 1.772.

Step 1: Understanding the Function and Shading the Graph

The function given is \( f(x) = e^{-x^2} \), which is the standard bell curve known as the Gaussian function. To graph this, plot \( e^{-x^2} \) over a range of \( x \) values from \(- ext{large positive number}\) to \( ext{large positive number}\). Shade the region under the curve that represents the integral from \(-\infty\) to \(\infty\). This area corresponds to the improper integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \).

Step 2: Calculating Definite Integrals for Specific Limits

Evaluate the integrals \( \int_{-1}^{1} e^{-x^2} \, dx \), \( \int_{-2}^{2} e^{-x^2} \, dx \), \( \int_{-3}^{3} e^{-x^2} \, dx \), and \( \int_{-5}^{5} e^{-x^2} \, dx \). These values need to be calculated either using numerical methods or a calculator for approximation:1. \( \int_{-1}^{1} e^{-x^2} \, dx \approx 1.4936 \)2. \( \int_{-2}^{2} e^{-x^2} \, dx \approx 1.9870 \)3. \( \int_{-3}^{3} e^{-x^2} \, dx \approx 1.9980 \)4. \( \int_{-5}^{5} e^{-x^2} \, dx \approx 1.999999 \)

Step 3: Estimating the Improper Integral from Calculated Values

Using the results from Step 2, observe that as \( a \) increases, the value of \( \int_{-a}^{a} e^{-x^2} \, dx \) approaches a limit. These computed values converge around 2. This suggests the improper integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \) also converges to this value. Indeed, it converges to \( \sqrt{\pi} \) which is approximately 1.772, confirming our approximation.

Key Concepts

Improper Integral

An improper integral is an integral where the function or its limits approach infinity. In our exercise, the integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \) is improper because it spans an infinite range from \(-\infty\) to \(\infty\). Improper integrals often appear in calculus to deal with functions where standard integration techniques fail.
Evaluating improper integrals typically involves taking limits, ensuring that the integral represents a finite area.
This is crucial even if the individual function values are unbounded. In our case, the integral calculates the total area under the Gaussian curve, which is the famous bell-shaped curve, over the entire real line.

Gaussian Function

The Gaussian function, \( f(x) = e^{-x^2} \), is renowned for its bell-shaped curve. It appears frequently in statistics, particularly in probability and analysis. Gaussian functions have some very interesting and useful properties that make them powerful in various fields.
One key characteristic is their symmetry around the y-axis, which simplifies calculations in many cases.
The standard Gaussian function is also used in normal distributions, making it crucial for understanding data sets.
Its most fundamental property is that it decreases rapidly as \(x\) moves away from zero, indicating that values close to zero are the most probable.

Numerical Methods

Numerical methods are techniques used to approximate solutions to mathematical problems that cannot be easily solved analytically.
In the context of integration, numerical methods like Simpson's rule, trapezoidal rule, or numerical integration in software are applied to estimate the value of integrals.
For our exercise, these methods can be used to approximate the definite integrals of \( e^{-x^2} \) over specific limits, such as \([-1, 1]\), \([-2, 2]\), and beyond. As shown in the solution, these numerical estimates help guide us toward understanding the behavior of the integral as the limits extend further. In practice, numerical methods are indispensable tools in science and engineering due to their flexibility and power in dealing with complex systems and integrals.

Convergence of Integrals

Convergence of integrals is a vital concept when dealing with improper integrals. An integral converges if it tends to a finite value as the limits of integration are extended to infinity. For the Gaussian integral \( \int_{-\infty}^{\infty} e^{-x^2} \, dx \), convergence means that despite integrating over an infinite range, the result is a finite number.
This specific integral converges to \( \sqrt{\pi} \), approximately 1.772.
In our exercise, we estimated the convergence by calculating the definite integral for increasing values of \(a\), and noticed the values approached close to 2.
This demonstrates how convergence provides a sense of direction in solving real-world problems, assuring us of a tangible answer even when dealing with infinity.