/*! 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 5 Explain why the integral is impr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 5

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{4} \frac{1}{\sqrt{x}} d x $$

Short Answer

Expert verified
The integral \(\int_{0}^{4} \frac{1}{\sqrt{x}} dx\) is termed improper due to the existence of a singularity at \(x = 0\). It is found to converge and has a value of \(4\) upon evaluation.

Step by step solution

01

Recognize the Improper Integral

The integral \(\int_{0}^{4} \frac{1}{\sqrt{x}} dx\) is considered improper because of the singularity at \(x = 0\), as \(f(x) = \frac{1}{\sqrt{x}}\) is undefined at \(x = 0\). Thus, the integral can be rewritten as \(\lim_{t \to 0+} \int_{t}^{4} \frac{1}{\sqrt{x}} dx).\)
02

Evaluate the Improper Integral

To determine whether the integral converges or diverges, and to find its value if it does converge, evaluate the limit of the integral as \(t\) goes to 0 from the right: \(\lim_{t \to 0+} \int_{t}^{4} \frac{1}{\sqrt{x}} dx).\) Using the antiderivative of \(\frac{1}{\sqrt{x}}\), which is \(2\sqrt{x}\), the definite integral from \(t\) to 4 can be found. This gives \(2\sqrt{4} - 2\sqrt{t}\).
03

Find the Limit and the Value of the Integral

Substitute \(t\) goes to 0 in the evaluated integral \(\lim_{t \to 0+} (2\sqrt{4} - 2\sqrt{t})\). This evaluates to \(4\), which happens to be the value of the integral. This means the integral converges, and its value is \(4.\)

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

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sinh a t $$

Find the integral. Use a computer algebra system to confirm your result. $$ \int\left(\tan ^{4} t-\sec ^{4} t\right) d t $$

(a) find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only by a constant. $$ \int \sec ^{2} x \tan x d x $$

Consider the integral \(\int_{0}^{\pi / 2} \frac{4}{1+(\tan x)^{n}} d x\) where \(n\) is a positive integer. (a) Is the integral improper? Explain. (b) Use a graphing utility to graph the integrand for \(n=2,4,\) \(8,\) and \(12 .\) (c) Use the graphs to approximate the integral as \(n \rightarrow \infty\). (d) Use a computer algebra system to evaluate the integral for the values of \(n\) in part (b). Make a conjecture about the value of the integral for any positive integer \(n\). Compare your results with your answer in part (c).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=t^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

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.