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Chapter 6: Problem 53

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?

Short Answer

Expert verified
Yes, the integral converges to 0.5.

Step by step solution

01

- 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).
02

- 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.
03

- 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.
04

- 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.
05

- 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.
06

- 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(\frac{1}{\text{蟺}}\text{e^{-x^2}}\) 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: \(\text{I} = \int_{0}^{+\text{鈭瀩} \frac{1}{鈰斚�} \text{e}^{-x^2}\text{ dx}.\)
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.

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Most popular questions from this chapter

Use the numeric integration feature of your calculator to compute $$ I(N)=\int_{1}^{N} \frac{\ln (x+1)}{x} d x $$ for \(N=10,100,1,000,10,000\). Based on your results, do you think the improper integral $$ \int_{1}^{+\infty} \frac{\ln (x+1)}{x} d x $$ converges? If so, to what value?

Approximate the given integral and estimate the error with the specified number of subintervals using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{1}^{2} \frac{e^{x}}{x} d x ; n=10 $$

SUBSCRIPTION GROWTH The publishers of a national magazine have found that the fraction of subscribers who remain subscribers for at least \(t\) years is \(f(t)=e^{-t / 10}\). Currently, the magazine has 20,000 subscribers and it is estimated that new subscriptions will be sold at the rate of 1,000 per year. Approximately how many subscribers will the magazine have in the long run?

DEMOGRAPHICS Demographic studies indicate that the fraction of the residents that will remain in a certain city for at least \(t\) years is \(f(t)=e^{-t / 20}\). The current population of the city is 100,000 , and it is estimated that \(t\) years from now, new people will be arriving at the rate of \(100 t\) people per year. If this estimate is correct, what will happen to the population of the city in the long run?

TOTAL COST FROM MARGINAL COST A manufacturer determines that the marginal cost of producing \(q\) units of a particular commodity is \(C^{\prime}(q)=\sqrt{q} e^{0.01 q}\) dollars per unit. a. Express the total cost of producing the first 8 units as a definite integral. b. Estimate the value of the total cost integral in part (a) using the trapezoidal rule with \(n=8\) subintervals.
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