Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.2.42

Capitolo 8, Problema 8.2.42

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.

Guida verificata passo dopo passo

First, identify the region bounded by the curves: \(y = \ln(x)\), \(y = 1\), the \(x\)-axis (\(y=0\)), and the \(y\)-axis (\(x=0\)). Since \(\ln(x)\) is defined for \(x > 0\), the region lies between \(x=1\) (where \(\ln(1)=0\)) and \(x=e\) (where \(\ln(e)=1\)).

Set up the volume integral using the method of washers (disks with holes) since the solid is generated by revolving the region around the \(x\)-axis. The volume element is \(\pi \left(R^2 - r^2\right) dx\), where \(R\) is the outer radius and \(r\) is the inner radius of the washer at position \(x\).

Determine the outer and inner radii of the washers. The outer radius corresponds to the line \(y=1\), so \(R = 1\). The inner radius corresponds to the curve \(y = \ln(x)\), so \(r = \ln(x)\).

Write the volume integral as \(V = \int_{x=1}^{x=e} \pi \left(1^2 - (\ln(x))^2\right) dx = \pi \int_{1}^{e} \left(1 - (\ln(x))^2\right) dx\).

To find the volume, evaluate the integral \(\int_{1}^{e} \left(1 - (\ln(x))^2\right) dx\). This may require integration by parts or substitution for the \(\int (\ln(x))^2 dx\) term.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes of slices or shells.

Video consigliato:

04:48

Finding Volume Using Disks

Disk/Washer Method

The disk/washer method calculates volume by slicing the solid perpendicular to the axis of rotation, forming circular disks or washers. The volume is found by integrating the area of these cross-sections along the axis of revolution.

Video consigliato:

06:30

Disk Method Using y-Axis

Natural Logarithm Function and Its Graph

Understanding the function f(x) = ln(x) is essential, including its domain (x > 0), range, and shape. Knowing how it interacts with the lines y = 1 and the coordinate axes helps define the bounded region to be revolved.

Video consigliato:

5:26

Graphs of Logarithmic Functions

Pratica correlata

Domanda del libro di testo

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

views

Domanda del libro di testo

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).

59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)

views

Domanda del libro di testo

3. Describe the method used to integrate sin³x.

views

Domanda del libro di testo

79. Tabular integration extended Refer to Exercise 77.

a. The following table shows the method of tabular integration applied to

∫ eˣ cos x dx.

Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.

b. Solve the equation in part (a) for ∫ eʳ cos z dz.

c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).