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Ch. 6 - Applications of Integration
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. 6 - Applications of IntegrationProblema 6.3.5c
Capitolo 6, Problema 6.3.5c

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). 


c. Write an integral for the volume of the solid.

Guida verificata passo dopo passo
1
Identify the region R bounded by the curve \(y = \sqrt{\cos x}\) and the x-axis on the interval \([0, \frac{\pi}{2}]\). This means the region lies between \(y = 0\) and \(y = \sqrt{\cos x}\) for \(x\) in \([0, \frac{\pi}{2}]\).
Since the solid is formed by revolving the region R about the x-axis, use the disk method to find the volume. The volume of a solid of revolution generated by revolving a curve \(y = f(x)\) about the x-axis from \(x = a\) to \(x = b\) is given by the integral \(V = \pi \int_a^b [f(x)]^2 \, dx\).
In this problem, the function is \(f(x) = \sqrt{\cos x}\), so the radius of each disk is \(\sqrt{\cos x}\). Squaring this radius gives the area of the cross-sectional disk as \([\sqrt{\cos x}]^2 = \cos x\).
Set up the integral for the volume using the limits of integration \(0\) to \(\frac{\pi}{2}\):
\[V = \pi \int_0^{\frac{\pi}{2}} \cos x \, dx.\]

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Region Bounded by a Curve and the x-axis

Understanding the region bounded by the curve y = √cos x and the x-axis on [0, π/2] is essential. This involves identifying the area under the curve from x = 0 to x = π/2, which forms the cross-sectional shape to be revolved around the x-axis.
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Finding Area Between Curves on a Given Interval

Volume of a Solid of Revolution Using the Disk Method

The disk method calculates the volume of a solid formed by revolving a region around the x-axis. The volume is found by integrating π times the square of the radius function (here, y = √cos x) with respect to x over the given interval.
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Finding Volume Using Disks

Setting up Definite Integrals for Volume

Writing the integral requires expressing the volume as an integral with proper limits and integrand. For this problem, the integral is from 0 to π/2 of π times (√cos x)² dx, simplifying to π∫₀^{π/2} cos x dx, which represents the volume of the solid.
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Percorso guidato
05:43
Definition of the Definite Integral
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