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Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.R.102
Capitolo 8, Problema 8.R.102

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.
102. About the y-axis

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Identify the region R bounded by the curve \(y = \ln(x)\) and the x-axis on the interval \([1, e]\). This means the region lies between \(x=1\) and \(x=e\), above the x-axis and below the curve \(y=\ln(x)\).
Since the solid is generated by revolving the region around the y-axis, consider using the method of cylindrical shells. The formula for the volume using shells is \(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\).
In this problem, the radius of a shell is the distance from the y-axis to a point \(x\), which is simply \(x\). The height of the shell is the value of the function \(y = \ln(x)\).
Set up the integral for the volume as \(V = \int_1^e 2\pi x \ln(x) \, dx\). This integral represents the sum of the volumes of all cylindrical shells from \(x=1\) to \(x=e\).
To find the volume, evaluate the integral \(\int_1^e x \ln(x) \, dx\) using integration by parts, then multiply the result by \(2\pi\).

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Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Understanding which method to apply depends on the axis of rotation and the shape of the region.
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Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed when a region is revolved around an axis. For rotation about the y-axis, vertical slices parallel to the axis create shells with radius equal to the x-value and height given by the function. The volume is found by integrating 2π(radius)(height) dx over the interval.
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Euler's Method

Natural Logarithm Function and Its Properties

The function y = ln(x) is defined for x > 0 and is the inverse of the exponential function. It is continuous and increasing on [1, e], with ln(1) = 0 and ln(e) = 1. Understanding its behavior helps determine the region bounded by y = ln(x) and the x-axis, which is essential for setting up the integral limits and expressions.
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Percorso guidato
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Properties of Functions
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Domanda del libro di testo

101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?

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109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by

v(t) = 10 * sin(3t).

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89–91. Comparison Test Determine whether the following integrals converge or diverge.

89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

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125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

40. ∫ (x² - 4)/(x + 4) dx

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Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.

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