Skip to main content
Ch. 6 - Applications of Definite Integrals
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 6 - Applications of Definite IntegralsProblema 6.PE.13b
Capitolo 6, Problema 6.PE.13b

Volumes
Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about
b. the line y = ―1

Guida verificata passo dopo passo
1
First, identify the region bounded by the curve \(y = x^{2} - 2x\) and the x-axis. To do this, find the points where the curve intersects the x-axis by solving \(x^{2} - 2x = 0\).
Next, since the solid is generated by revolving the region about the line \(y = -1\), determine the radius of the typical disk or washer. The radius will be the vertical distance from the curve to the line \(y = -1\). Express this radius as \(R(x) = (y - (-1)) = (x^{2} - 2x + 1)\).
Because the region is bounded below by the x-axis (\(y=0\)), the inner radius of the washer is the distance from \(y=0\) to \(y=-1\), which is \(r = 1\). So, the volume element is a washer with outer radius \(R(x)\) and inner radius \(r\).
Set up the volume integral using the washer method: \(V = \pi \int_{a}^{b} \left[R(x)^{2} - r^{2}\right] \, dx\), where \(a\) and \(b\) are the x-values found in step 1.
Finally, expand the integrand, simplify, and prepare to integrate with respect to \(x\) over the interval \([a, b]\) to find the volume.

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:
10m

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 a line. The volume is typically calculated using integral methods such as the disk, washer, or shell method, depending on the axis of rotation and the shape of the region.
Video consigliato:
04:48
Finding Volume Using Disks

Washer Method

The washer method is used when the solid has a hollow center, created by revolving a region around a line that does not coincide with the axis of the curve. It involves subtracting the volume of the inner radius from the outer radius, integrating the difference of squared radii times π.
Video consigliato:
07:33
Euler's Method

Adjusting for Axis of Rotation (Line y = -1)

When revolving around a line other than the x-axis or y-axis, distances (radii) must be measured relative to that line. For y = -1, the radius is the vertical distance between the curve and y = -1, requiring careful adjustment of the function values before applying the volume formula.
Video consigliato:
06:30
Disk Method Using y-Axis
Pratica correlata
Domanda del libro di testo

Work


Assume that a spring does not follow Hooke’s Law. Instead, the force required to stretch the spring x ft from its natural length is ƒ(𝓍) = 10𝓍³/² lb . How much work does it take to

a. stretch the spring 4 ft from its natural length?

21
views
Domanda del libro di testo

Volumes

Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about

a. the x-axis

29
views
Domanda del libro di testo

Volumes

Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about

c. the line x = 1

60
views
Domanda del libro di testo

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = π/4 and x = 5π/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.

36
views
Domanda del libro di testo

Find the lengths of the curves in Exercises 19–22.

y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32

24
views
Domanda del libro di testo

Volumes

Volume of a solid sphere hole A round hole of radius √3 ft is bored through the center of a solid sphere of radius 2 ft. Find the volume of material removed from the sphere.

44
views