/*! 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 29 Determine whether the improper i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 29

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$

Short Answer

Expert verified
The result of the evaluation depends on the calculation. The final result will be given by the sum of the two evaluated integrals if they form a finite number. If not, then the integral diverges. This is confirmed by checking the integral using a graphing utility.

Step by step solution

01

Identify the function

This is an integral of the function \(f(x) = \frac{1}{\sqrt[3]{x-1}}\) from 0 to 2.
02

Break the integral at the points of discontinuity

Since x = 1 is a point of discontinuity, we can break the integral into two, from 0 to 1 and from 1 to 2. So, \(\int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x = \int_{0}^{1} \frac{1}{\sqrt[3]{x-1}} d x + \int_{1}^{2} \frac{1}{\sqrt[3]{x-1}} d x\)
03

Solve the integral using limit

Now we take the limit approach to solve this integral. For the first integral, we can rewrite as \(\lim_{{t \to 1^-}} \int_{0}^{t} \frac{1}{\sqrt[3]{x-1}} d x\). For the second integral, we can rewrite as \(\lim_{{t \to 1^+}} \int_{t}^{2} \frac{1}{\sqrt[3]{x-1}} d x\)
04

Evaluate the integral

Now evaluate these integrals and add them. If they form a finite number then the integral converges and the finite value would be that value. If the result is undefined or infinite, the integral would diverge.
05

Check with graphing utility

Plug the integral into a graphing utility to confirm your results.

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

Lumber Use The table shows the amounts of lumber used for residential upkeep and improvements (in billions of board-feet per year) for the years 1997 through 2005 . $$ \begin{array}{|c|c|c|c|c|c|}\hline \text { Year } & {1997} & {1998} & {1999} & {2000} & {2001} \\ \hline \text { Amount } & {15.1} & {14.7} & {15.1} & {16.4} & {17.0} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline \text { Year } & {2002} & {2003} & {2004} & {2005} \\ \hline \text { Amount } & {17.8} & {18.3} & {20.0} & {20.6} \\\ \hline\end{array} $$ (a) Use Simpson's Rule to estimate the average number of board-feet (in billions) used per year over the time period. (b) A model for the data is $$L=6.613+0.93 t+2095.7 e^{-t}, \quad 7 \leq t \leq 15$$ where \(L\) is the amount of lumber used and \(t\) is the year, with \(t=7\) corresponding to \(1997 .\) Use integration to find the average number of board- feet (in billions) used per year over the time period. (c) Compare the results of parts (a) and (b).

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{-1} \frac{1}{x^{2}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{1-x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{5}{e^{2 x}} d x $$

Determine the amount of money required to set up a charitable endowment that pays the amount \(P\) each year indefinitely for the annual interest rate \(r\) compounded continuously. $$ P=\$ 5000, r=7.5 \% $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision 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.