Problem 53 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Use the numeric integration feature of your calculator to compute $$ I(N)=\int_{0}^{N} \frac{1}{\sqrt{\pi}} e^{-x^{2}} d x $$ for \(N=1,10,50\). Based on your results, do you think the improper integral $$ \int_{0}^{+\infty} \frac{1}{\sqrt{\pi}} e^{-x^{2}} d x $$ converges? If so, to what value?

Step-by-Step Solution

Answer

Yes, the integral converges to 0.5.

Step 1 - Set Up the Integral

Understand that you need to compute the value of the definite integral \ \[ I(N)=\int_{0}^{N} \frac{1}{\sqrt{\pi}} e^{-x^{2}} dx \] \ for each given value of N (1, 10, 50).

Step 2 - Use Calculator for N=1

Use your calculator's numeric integration function to compute \ \[ I(1)=\int_{0}^{1} \frac{1}{\sqrt{\pi}} e^{-x^{2}} dx. \] \ Record the result.

Step 3 - Use Calculator for N=10

Use your calculator's numeric integration function to compute \ \[ I(10)=\int_{0}^{10} \frac{1}{\sqrt{\pi}} e^{-x^{2}} dx. \] \ Record the result.

Step 4 - Use Calculator for N=50

Use your calculator's numeric integration function to compute \ \[ I(50)=\int_{0}^{50} \frac{1}{\sqrt{\pi}} e^{-x^{2}} dx. \] \ Record the result.

Step 5 - Analyze the Results

Compare the values of the integrals for N=1, N=10, and N=50. Notice if the results are approaching a certain value as N increases.

Step 6 - Determine Convergence

Based on the values obtained, determine if the improper integral \ \[ \int_{0}^{+\infty} \frac{1}{\sqrt{\pi}} e^{-x^{2}} dx \] \ converges. Identify to what value it converges if applicable.

Key Concepts

Definite Integral

A definite integral is a way to find the area under a curve between two specific points. In the exercise, we are given a function \(\frac{1}{\text{π}}\text{e^{-x^2}}\) and need to compute the definite integral from 0 to a specific upper limit N. This area gives us an idea of how much 'space' lies beneath the curve.

Improper Integral

An improper integral extends the concept of a definite integral. In this case, one or both of the limits of integration are infinite. The exercise asks us to evaluate an improper integral where the upper limit approaches infinity: \(\text{I} = \int_{0}^{+\text{∞}} \frac{1}{⋔π} \text{e}^{-x^2}\text{ dx}.\)

Gaussian Function

The Gaussian function \(e^{-x^2}\), is essential in probability and statistics, especially in the context of the normal distribution. In our integral, it determines the integrand's shape, making the value of our improper integral equivalent to the area under the standard normal distribution curve.