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Ch. 8 - Techniques of Integration
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. 8 - Techniques of IntegrationProblema 8.8.90b
Capitolo 8, Problema 8.8.90b

90. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / √x, y = 0, x = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (i) about the x-axis

Guida verificata passo dopo passo
1
Identify the region bounded by the curves: \(y = \frac{1}{\sqrt{x}}\), \(y = 0\), \(x = 0\), and \(x = 1\). This region lies in the first quadrant between \(x=0\) and \(x=1\) under the curve \(y = \frac{1}{\sqrt{x}}\).
Since the region is revolved about the x-axis, use the disk method to find the volume. The volume \(V\) is given by the integral \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the function representing the radius of the disks.
Here, the radius of each disk is \(y = \frac{1}{\sqrt{x}}\). So, the volume integral becomes \(V = \pi \int_{0}^{1} \left( \frac{1}{\sqrt{x}} \right)^{2} \, dx\).
Simplify the integrand: \(\left( \frac{1}{\sqrt{x}} \right)^{2} = \frac{1}{x}\). So the integral is \(V = \pi \int_{0}^{1} \frac{1}{x} \, dx\).
Evaluate the integral \(\int_{0}^{1} \frac{1}{x} \, dx\). Note that this integral is improper at \(x=0\), so consider it as a limit: \(\lim_{t \to 0^{+}} \int_{t}^{1} \frac{1}{x} \, dx\). Set up this limit to analyze the behavior of the volume.

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Definite Integrals for Area and Volume

Definite integrals calculate the accumulation of quantities, such as area under a curve or volume of a solid. In this problem, the integral bounds are from 0 to 1, and the integral will be used to find the volume generated by revolving a region around an axis.
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Definition of the Definite Integral

Volume of Solids of Revolution (Disk/Washer Method)

When a region is revolved around an axis, the volume of the resulting solid can be found using the disk or washer method. This involves integrating the area of circular cross-sections perpendicular to the axis of revolution, typically using the formula V = π∫[R(x)]² dx.
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Finding Volume Using Disks

Understanding the Region Bounded by Curves

Identifying the region bounded by y = 1/√x, y = 0, x = 0, and x = 1 is crucial. This region lies in the first quadrant and defines the shape to be revolved. Correctly interpreting these boundaries ensures the integral is set up with proper limits and functions.
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Finding Area When Bounds Are Not Given
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