Domanda del libro di testo
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
b. Show that the inner and outer surfaces of the solid have infinite area.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.8.89c
89. Consider the infinite region in the first quadrant bounded by the graphs of
y = 1 / x², y = 0, and x = 1.
b. Find the volume of the solid formed by revolving the region (ii) about the y-axis.
Guida verificata passo dopo passo1
Identify the region to be revolved: it is bounded by the curve \(y = \frac{1}{x^{2}}\), the line \(y = 0\), and the vertical line \(x = 1\), all in the first quadrant.
Since the region is revolved about the y-axis, consider using the method of cylindrical shells. The shell radius is the distance from the y-axis, which is \(x\), and the shell height is the vertical distance between the curve and the x-axis, which is \(\frac{1}{x^{2}}\).
Set up the volume integral using the shell method formula: \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\). Here, \(a = 1\) and \(b \to \infty\) because the region extends infinitely to the right.
Write the integral explicitly: \(V = 2\pi \int_{1}^{\infty} x \cdot \frac{1}{x^{2}} \, dx = 2\pi \int_{1}^{\infty} \frac{1}{x} \, dx\).
Evaluate the improper integral by taking the limit as the upper bound approaches infinity, and analyze whether the volume converges or diverges.
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Setting up the Region for Integration
Understanding the boundaries of the region is essential. Here, the region is in the first quadrant bounded by y = 1/x², y = 0, and x = 1. Identifying these limits helps determine the interval and shape of the area to be revolved.
Video consigliato:
Percorso guidato
07:45
Area of Polar Regions
Method of Cylindrical Shells for Volume
When revolving a region around the y-axis, the cylindrical shells method is often used. It involves integrating the volume of thin cylindrical shells with radius x, height given by the function, and thickness dx, which simplifies volume calculation.
Video consigliato:
04:48Finding Volume Using Disks
Improper Integrals and Convergence
Since the region extends infinitely in the y-direction, the volume integral is improper. Evaluating whether the integral converges (has a finite value) is crucial to ensure the volume is well-defined and to compute it correctly.
Video consigliato:
Percorso guidato
11:11Improper Integrals: Infinite Intervals
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Domanda del libro di testo
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 (ii) about the y-axis.
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