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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 6.3.26
Capitolo 6, Problema 6.3.26

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x,y=x+2,x=0, and x=4 ; about the x-axis
Graph showing region bounded by y = x, y = x + 2, x = 0, and x = 4 shaded between two lines.

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Identify the region R bounded by the curves y = x, y = x + 2, x = 0, and x = 4. This region lies between the two lines y = x and y = x + 2, from x = 0 to x = 4.
Since the region is revolved about the x-axis, use the method of washers to find the volume. The volume element dV is given by the difference of the areas of the outer and inner disks times the thickness dx.
The outer radius R(x) is the distance from the x-axis to the upper curve y = x + 2, so R(x) = x + 2. The inner radius r(x) is the distance from the x-axis to the lower curve y = x, so r(x) = x.
Write the volume integral as: \(V = \pi \int_0^4 \left[ (R(x))^2 - (r(x))^2 \right] \, dx = \pi \int_0^4 \left[ (x+2)^2 - x^2 \right] \, dx\).
Expand the integrand, simplify, and then integrate with respect to x over the interval [0,4] to find the volume of the solid.

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Region Bounded by Curves

The region R is defined by the lines y = x, y = x + 2, x = 0, and x = 4. Understanding how these curves intersect and enclose an area is essential for setting up the integral. The region lies between two parallel lines y = x and y = x + 2, bounded vertically by x = 0 and x = 4.
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Finding Area When Bounds Are Not Given

Volume of Solids of Revolution

When a region is revolved around an axis, it generates a 3D solid. The volume of this solid can be found using methods like the disk/washer or shell method. Here, revolving around the x-axis suggests using washers formed by the vertical distance between the curves.
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Finding Volume Using Disks

Washer Method for Volume Calculation

The washer method involves slicing the solid perpendicular to the axis of revolution, creating washers with an outer and inner radius. The volume is the integral of the area of these washers along the axis. For this problem, the outer radius is y = x + 2 and the inner radius is y = x, integrated from x = 0 to x = 4.
Video consigliato:
04:48
Finding Volume Using Disks
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