Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.2.69
Capitolo 8, Problema 8.2.69

69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?

Guida verificata passo dopo passo
1
First, identify the region R bounded by the curve \(y = \sin x\) and the x-axis on the interval \([0, \pi]\). This means the region lies between \(y = 0\) and \(y = \sin x\) for \(x\) in \([0, \pi]\).
To find the volume when R is revolved about the x-axis, use the disk method. The volume \(V_x\) is given by the integral: \(V_x = \pi \int_0^{\pi} (\sin x)^2 \, dx\) because the radius of each disk is \(\sin x\).
To find the volume when R is revolved about the y-axis, use the shell method. The volume \(V_y\) is given by the integral: \(V_y = 2\pi \int_0^{\pi} x \cdot \sin x \, dx\) where \(x\) is the radius of the shell and \(\sin x\) is the height.
Set up both integrals explicitly and prepare to evaluate them (though we won't compute the final values here). This involves recognizing the integrals: \(\int_0^{\pi} (\sin x)^2 \, dx\) and \(\int_0^{\pi} x \sin x \, dx\).
After evaluating both integrals, compare the numerical values of \(V_x\) and \(V_y\) to determine which volume is greater.

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:
6m

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 generated when a region is revolved around an axis. Common methods include the disk/washer method for revolution about the x-axis and the shell method for revolution about the y-axis. Understanding these methods helps set up the correct integrals for volume calculation.
Video consigliato:
04:48
Finding Volume Using Disks

Disk/Washer Method

Used when revolving a region around the x-axis, this method slices the solid perpendicular to the axis of revolution, creating circular disks or washers. The volume is found by integrating the area of these cross-sectional disks along the interval of interest.
Video consigliato:
06:30
Disk Method Using y-Axis

Shell Method

This method is useful for revolving a region around the y-axis. It involves slicing the region parallel to the axis of revolution, forming cylindrical shells. The volume is calculated by integrating the lateral surface area of these shells over the given interval.
Video consigliato:
07:33
Euler's Method
Pratica correlata
Domanda del libro di testo

7–84. Evaluate the following integrals.

45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

52
views
Domanda del libro di testo

Evaluate the following integrals.

∫ (1 - cosx)/(1 + cosx) dx

228
views
Domanda del libro di testo

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

38
views
Domanda del libro di testo

9–61. Trigonometric integrals Evaluate the following integrals.

57. ∫ from 0 to π of (1 - cos2x)³ᐟ² dx

78
views
Domanda del libro di testo

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

34. ∫ dx / (x(x¹⁰ + 1))

46
views
Domanda del libro di testo

Evaluate the following integrals.

∫ eˣ/(e²ˣ + 2eˣ + 17) dx

35
views