Domanda del libro di testo
Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
c. 2π ≤ x ≤ 3π.
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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 8.8.89b
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 (i) about the x-axis.
Guida verificata passo dopo passo1
Identify the region to be revolved: it is bounded by the curve \(y = \frac{1}{x^2}\), the x-axis (\(y=0\)), and the vertical line \(x=1\), in the first quadrant. Since the region is in the first quadrant, \(x\) ranges from 1 to infinity.
Set up the volume integral using the disk method because the region is revolved around the x-axis. The radius of each disk is the distance from the x-axis to the curve, which is \(r(x) = \frac{1}{x^2}\).
Write the volume integral as \(V = \pi \int_{1}^{\infty} \left(r(x)\right)^2 \, dx = \pi \int_{1}^{\infty} \left(\frac{1}{x^2}\right)^2 \, dx\).
Simplify the integrand: \(\left(\frac{1}{x^2}\right)^2 = \frac{1}{x^4}\), so the integral becomes \(V = \pi \int_{1}^{\infty} \frac{1}{x^4} \, dx\).
Evaluate the improper integral by finding the antiderivative of \(x^{-4}\), then take the limit as the upper bound approaches infinity to determine 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 solids. In volume problems, integrals sum infinitesimal cross-sectional areas along an axis to find total volume. Setting correct limits and integrand expressions is essential.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Method of Disks/Washers for Volumes of Revolution
The disk/washer method finds volumes by slicing the solid perpendicular to the axis of revolution. Each slice forms a disk or washer whose area is integrated along the axis. For revolution about the x-axis, the radius is the function value, and volume is π∫[radius]^2 dx.
Video consigliato:
04:48Finding Volume Using Disks
Understanding the Region and Boundaries
Identifying the region bounded by y = 1/x², y = 0, and x = 1 in the first quadrant is crucial. This defines the limits of integration and the shape being revolved. Recognizing that the region extends from x=1 to infinity helps set proper integral bounds for volume calculation.
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Percorso guidato
07:45Area of Polar Regions
Pratica correlata
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Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
c. u = arctan x
What is the value of the integral?
25
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Domanda del libro di testo
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.
19
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Domanda del libro di testo
Consider the region bounded by the graphs of y = sin⁻¹(x), y = 0, and x = 1/2.
b. Find the centroid of the region.
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Domanda del libro di testo
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
b. 3π/2 ≤ x ≤ 5π/2.
33
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Domanda del libro di testo
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 2 to 4 of 1/(s - 1)² ds
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