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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.R.104
Chapter 8, Problem 8.R.104

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.
104. About the line y = 1

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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 \(y = 0\) and \(y = \ln(x)\) for \(x\) from 1 to \(e\).
Since the solid is generated by revolving the region around the line \(y = 1\), consider the distance from the curve and the x-axis to this line. The radius of a typical disk or washer will be the vertical distance from \(y = 1\) to the curve or axis.
Set up the volume integral using the washer method. The outer radius \(R_{outer}\) is the distance from \(y = 1\) to the x-axis (\(y=0\)), which is \(1 - 0 = 1\). The inner radius \(R_{inner}\) is the distance from \(y = 1\) to the curve \(y = \ln(x)\), which is \(1 - \ln(x)\).
Write the volume integral as \(V = \pi \int_1^{e} \left(R_{outer}^2 - R_{inner}^2\right) \, dx = \pi \int_1^{e} \left(1^2 - (1 - \ln(x))^2\right) \, dx\).
Expand the integrand and simplify it before integrating. Then, integrate term-by-term with respect to \(x\) over the interval \([1, e]\) to find the volume.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around a line. The volume is typically calculated using integral methods such as the disk, washer, or shell method, depending on the axis of rotation and the shape of the region.
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Finding Volume Using Disks

Washer Method

The washer method is used when the solid has a hollow center, created by revolving a region around a line that does not coincide with the axis of the region. It involves subtracting the volume of the inner radius from the outer radius, integrating π(outer radius² - inner radius²) over the interval.
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Euler's Method

Natural Logarithm Function and Its Graph

Understanding the function y = ln(x) is essential, including its behavior on the interval [1, e]. The curve lies above the x-axis here, and knowing its shape helps determine the radii for the washers when revolving around the line y = 1.
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Graphs of Logarithmic Functions
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