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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 6.3.24
Capitolo 6, Problema 6.3.24

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 and y=4√x; about the x-axis
Graph showing region bounded by y = x and y = 4√x, shaded area revolved around x-axis forming a solid.

Guida verificata passo dopo passo
1
Identify the region R bounded by the curves \(y = x\) and \(y = 4\sqrt{x}\), and the axis of revolution, which is the x-axis.
Find the points of intersection of the curves by setting \(x = 4\sqrt{x}\). Solve for \(x\) to determine the limits of integration.
Since the region is revolved around the x-axis, use the washer method to find the volume. The volume element is given by \(\pi \int_a^b \left(R(x)^2 - r(x)^2\right) \, dx\), where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius.
Determine which curve is the outer radius and which is the inner radius relative to the x-axis. Here, the outer radius is the curve farther from the x-axis, and the inner radius is the closer curve. Express these radii as functions of \(x\).
Set up the integral for the volume using the limits of integration found in step 2 and the radii from step 4. The integral will be \(V = \pi \int_a^b \left( (4\sqrt{x})^2 - (x)^2 \right) \, dx\). Evaluate this integral to find the volume.

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Finding the Region of Integration

To find the volume of the solid, first identify the region bounded by the curves y = x and y = 4√x. Determine their points of intersection by solving x = 4√x, which sets the limits of integration along the x-axis. This step is crucial to correctly define the interval over which the volume will be calculated.
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Area of Polar Regions

Volume of Solids of Revolution Using the Washer Method

When a region is revolved around the x-axis, the volume can be found using the washer method. This involves integrating the difference of the squares of the outer and inner radii (functions of x) multiplied by π over the interval. Here, the outer radius is the upper curve and the inner radius is the lower curve relative to the axis of rotation.
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Finding Volume Using Disks

Setting up and Evaluating the Definite Integral

After determining the limits and radii, set up the definite integral for volume: V = π∫[a to b] (R(x))^2 - (r(x))^2 dx. Carefully substitute the expressions for R(x) and r(x) from the given curves, then evaluate the integral using appropriate techniques such as substitution or power rule to find the exact volume.
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Definition of the Definite Integral
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