Skip to main content
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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

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.
Recommended video:
04:48
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.
Recommended video:
07:33
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.
Recommended video:
5:26
Graphs of Logarithmic Functions
Related Practice
Textbook Question

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).

Textbook Question

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

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

1
views
Textbook Question

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

25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)

Textbook Question

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

1
views
Textbook Question

119. {Use of Tech} Comparing volumes Let R be the region bounded by y = ln(x), the x-axis, and the line x = a, where a > 1.

b. Find the volume V₂(a) of the solid generated when R is revolved about the y-axis (as a function of a).

Textbook Question

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

57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx