/*! 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 24 Find the volume generated by rot... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 24

Find the volume generated by rotating about the \(x\) -axis the regions bounded by the graphs of each set of equations. $$y=\sqrt{x e^{-x}}, x=1, x=2$$

Short Answer

Expert verified
\( \pi (-2e^{-2} + e^{-1}) \)

Step by step solution

01

- Understand the Problem

We need to find the volume generated by rotating the given region about the x-axis. The equations are given as: 1. \(y = \sqrt{x e^{-x}}\)2. \(x = 1\)3. \(x = 2\)
02

- Set Up the Integral for Volume

The volume of a solid of revolution about the x-axis is given by the formula: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]Here, \(f(x) = \sqrt{x e^{-x}}\). The limits of integration, \(a\) and \(b\), are 1 and 2, respectively.
03

- Express the Integrand

First, simplify the integrand. Since \(f(x) = \sqrt{x e^{-x}}\),\[ [f(x)]^2 = (\sqrt{x e^{-x}})^2 = x e^{-x} \]So, the integral becomes:\[ V = \pi \int_{1}^{2} x e^{-x} \, dx \]
04

- Integrate the Function

To integrate \(x e^{-x}\), we use integration by parts. Let \(u = x\) and \(dv = e^{-x} dx\). Then \(du = dx\) and \(v = -e^{-x}\). Using integration by parts formula \(\int u dv = uv - \int v du\), we get:\[ \int x e^{-x} \, dx = -x e^{-x} - \int -e^{-x} \, dx \]Simplify and integrate:\[ -x e^{-x} + e^{-x} \]Evaluate this from 1 to 2.
05

- Evaluate the Definite Integral

Substitute the limits into the antiderivative:\[ \left[ -x e^{-x} + e^{-x} \right]_{1}^{2} = \left( -2 e^{-2} + e^{-2} \right) - \left( -1 e^{-1} + e^{-1} \right) \]Simplify:\[ = -e^{-2} - (-e^{-1}) + (e^{-1} - e^{-2}) \]Combine like terms:\[ = -2e^{-2} + e^{-1} \]
06

- Calculate the Final Volume

Multiply the result by \(\pi\) to get the volume:\[ V = \pi (-2e^{-2} + e^{-1}) \]

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.

integration by parts
Integration by parts is a crucial technique for solving integrals involving the product of two functions. The method is derived from the product rule for differentiation and is represented by the formula \[\int u \, dv = uv - \int v \, du \]. This formula allows us to break down complex integrals into simpler ones.

Understanding each part of the formula is very important:
  • \

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

The number of loaves of bread, \(N\), baked each day by Fireside Bakers is normally distributed with mean 1000 and standard deviation \(50 .\) The bakery pays bonuses to its employees on those days when at least 1100 loaves are baked. What percentage of days will the bakery have to pay a bonus?

Find the demand function \(q=D(x)\), given each set of elasticity conditions. $$E(x)=\frac{4}{x} ; \quad q=e \text { when } x=4$$

For every 100,000 males in the United States of any age \(x\) between 1 and \(85,\) the number that died in 2003 can be approximated by the function $$f(x)=17.359 e^{0.067 x}$$(Source: Centers for Disease Control and Prevention.) (GRAPH CAN'T COPY) a) Find a value \(k\) to make the function \(k f(x)\) a probability density function. b) Use your answer from part (a) to approximate the probability that a male aged 1 through 85 who died in 2003 was between 20 and 30 years old.

Life science: drug dosage. Suppose that an oral dose of a drug is taken. Over time, the drug is assimilated in the body and excreted through the urine. The total amount of the drug that has passed through the body in time T is given by $$ \int_{0}^{T} E(t) d t$$ where \(E(t)\) is the rate of excretion of the drug. A typical rate-ofexcretion function is \(E(t)=t e^{-k t},\) where \(k>0\) and \(t\) is the time, in hours. Use this information for Exercises 49 and \(50 .\) Find \(\int_{0}^{\infty} E(t) d t,\) and interpret the answer. That is, what does the integral represent?

In a psychology experiment, the time \(t\) in seconds, that it takes a rat to learn its way through a maze is an exponentially distributed random variable with the probability density function $$f(t)=0.02 e^{-0.02 t}, \quad 0 \leq t<\infty$$. Find the probability that a rat will learn its way through a maze in 150 sec or less.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.