/*! 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 12 Determine whether the improper i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 12

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} d x $$

Short Answer

Expert verified
The improper integral \(\int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} dx\) diverges.

Step by step solution

01

Define the Improper Integral as Limit

Set the improper integral as the limit of a proper integral. Here, instead of the upper limit of integration being infinite, replace it with \(t\), where \(t\) will later approach infinity. Hence, \(\int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} dx = \lim_{t \to \infty} \int_{1}^{t} \frac{1}{\sqrt[3]{x}} dx\).
02

Evaluate the Integral

Next, compute the integral without considering the limit: \(\int_{1}^{t} \frac{1}{\sqrt[3]{x}} dx\). This is a basic power rule problem in integral calculus, where the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). The integral becomes \(3 (t^{\frac{2}{3}} - 1)\).
03

Applying the Limit

Now, apply the limit: \(\lim_{t \to \infty} 3 (t^{\frac{2}{3}} - 1)\). As \(t\) approaches infinity, \(t^{\frac{2}{3}}\) also approaches infinity. Thus, the limit is infinite.
04

Determining Convergence or Divergence

When the value of the limit is infinite, the integral diverges. Therefore, the improper integral \(\int_{1}^{\infty} \frac{1}{\sqrt[3]{x}} dx\) diverges, and there exists no finite area under the curve from 1 to infinity for the function \(f(x) = \frac{1}{\sqrt[3]{x}}\).

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!

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

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x \sqrt{x+4} d x $$

Profit The net profits \(P\) (in billions of dollars per year) for The Hershey Company from 2002 through 2005 can be modeled by \(P=\sqrt{0.00645 t^{2}+0.1673}, \quad 2 \leq t \leq 5\) where \(t\) is time in years, with \(t=2\) corresponding to 2002 . Find the average net profit over that time period. (Source: The Hershey Co.)

Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} x^{3} d x $$

Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$

Use the definite integral below to find the required arc length. If \(f\) has a continuous derivative, then the arc length of \(f\) between the points \((a, f(a))\) and \((b, f(b))\) is \(\int_{b}^{a} \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x\) Arc Length A fleeing hare leaves its burrow \((0,0)\) and moves due north (up the \(y\) -axis). At the same time, a pursuing lynx leaves from 1 yard east of the burrow \((1,0)\) and always moves toward the fleeing hare (see figure). If the lynx's speed is twice that of the hare's, the equation of the lynx's path is \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) Find the distance traveled by the lynx by integrating over the interval \([0,1]\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.