/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Determine whether each integral ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 11

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{-\infty}^{\infty} x e^{-x^{2}} d x\)

Short Answer

Expert verified
The integral is convergent and evaluates to 0.

Step by step solution

01

Setting up the Integral

The given integral is \( \int_{-\infty}^{\infty} x e^{-x^{2}} \, dx \). We need to determine whether this integral is convergent or divergent.
02

Analyze the Symmetry

Notice that the integrand \( x e^{-x^2} \) is an odd function, meaning that \( f(-x) = -f(x) \). This is because the term \( x \) changes sign while \( e^{-x^{2}} \) remains unchanged when \( x \) is replaced by \( -x \).
03

Using Properties of Odd Functions and Integrals

Integrals over symmetric limits \([-a, a]\) for odd functions are zero. Thus,\[ \int_{-\infty}^{\infty} x e^{-x^{2}} \, dx = \int_{-a}^{a} x e^{-x^{2}} \, dx = 0 \] for all \( a \), and letting \( a \to \infty \) shows:\[ \int_{-\infty}^{\infty} x e^{-x^{2}} \, dx = 0. \]
04

Conclusion on Convergence

Since the integral evaluates to 0, it is convergent. This is due to the symmetry and properties of odd functions over symmetric intervals.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Convergence and Divergence
When we talk about convergence and divergence, we're exploring whether an integral or series "settles" to a specific value as you evaluate it over certain limits. For example, when examining a definite integral like \( \int_{-\infty}^{\infty} x e^{-x^{2}} \, dx \), we want to see if it gives a real number as a result.

If the integral sums to a finite value, it is said to be convergent. Conversely, if it tends to infinity or doesn't settle to a single value, it is divergent.
Here are a few properties to keep in mind:
  • An integral is convergent if it results in a finite number when evaluated over infinite limits.
  • Divergent integrals either grow without bound or oscillate without approaching a single value.
Thus, determining convergence is essential to ensure that the integral or series behaves nicely and yields meaningful results.
Integration of Odd Functions
Odd functions have a distinct property that makes them stand out in integrals, especially over symmetric intervals. These functions satisfy the condition \( f(-x) = -f(x) \), effectively mirroring about the origin on the graph.

The function \( x e^{-x^2} \) in our integral falls into the category of odd functions because the change in the sign of \( x \) results in the entire function changing sign.
  • The significance of integrating odd functions over symmetric intervals, such as from \(-a\) to \(a\), is that their integrals become zero. This is because the positive and negative parts of the function cancel each other out perfectly.
  • Therefore, if you integrate an odd function from \(-\infty\) to \(\infty\), you can conclude that the integral converges to zero without performing the actual integration steps.
Understanding this property significantly simplifies complex integrations and is a powerful tool in calculus.
Properties of Symmetric Intervals
Symmetric intervals are ranges that are equidistant from zero, such as \([-a, a]\). These intervals possess special traits when dealing with certain functions, particularly odd functions as seen in our example of \( x e^{-x^2} \).

Here's why symmetric intervals are essential:
  • When integrating functions over symmetric intervals, values from the negative side can offset those from the positive side. This perfectly applies to odd functions.
  • The mathematical beauty of symmetric intervals lies in their ability to simplify calculations by reducing problems of undefined behavior over infinite limits into simpler, often null results.
Using these properties, we were able to see that an otherwise complex integral like the one given can resolve quickly to zero without endless computations just by recognizing its symmetry and the odd nature of the function in question.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(f\) is continuous on \(\mathbb{R},\) prove that $$\int_{a}^{b} f(x+c) d x=\int_{a+c}^{b+c} f(x) d x$$ For the case where \(f(x) \geqslant 0,\) draw a diagram to interpret this equation geometrically as an equality of areas.

$$\begin{array}{l}{\text { Sterile insect technique One method of slowing the }} \\ {\text { growth of an insect population without using pesticides is }} \\\ {\text { to introduce into the population a number of sterile males }} \\\ {\text { that mate with fertile females but produce no offspring. }}\end{array}$$. $$ \begin{array}{l}{\text { (The photo shows a screw-worm fly, the first pest effec- }} \\ {\text { tively eliminated from a region by this method.) Let } P} \\ {\text { represent the number of female insects in a population and }} \\\ {S \text { the number of sterile males introduced each generation. }}\end{array}$$. $$\begin{array}{c}{\text { Let } r \text { be the per capita rate of production of females by }} \\ {\text { females, provided their chosen mate is not sterile. Then the }} \\ {\text { female population is related to time } t \text { by }} \\ {t=\int \frac{P+S}{P[(r-1) P-S]} d P}\end{array}$$ $$\begin{array}{l}{\text { Suppose an insect population with } 10,000 \text { females grows at }} \\ {\text { a rate of } r=1.1 \text { and } 900 \text { sterile males are added. Evaluate }} \\ {\text { the integral to give an equation relating the female popula- }} \\ {\text { tion to time. (Note that the resulting equation can't be }} \\ {\text { solved explicitly for } P .}\end{array}$$

Growing degree days The rate of development of many plant species depends on the temperature of the environment in such a way that maturity is reached only after a certain number of "degree-days." Suppose that temperature \(T\) as a function of time \(t\) is given by $$T(t)=15\left(1+\sin \frac{2 \pi t}{60}\right) \quad 0 \leqslant t \leqslant 60$$ where \(t\) is measured in days. If maturity is reached on day\(t=20,\) what is the number of degree-days required? \(\left[\) Hint: What are the units for \(\int_{0}^{20} T(t) d t ?\)]$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{2 \pi}^{\infty} \sin \theta d \theta\)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \(\int_{e}^{\infty} \frac{1}{x(\ln x)^{3}} d x\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.