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 8.9.75
Chapter 8, Problem 8.9.75

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.

Verified step by step guidance
1
Identify the region to be revolved: The region is bounded by the curve \(f(x) = (4 - x)^{-\frac{1}{3}}\), the x-axis, and the interval \([0,4)\) on the x-axis.
Since the solid is revolved about the y-axis, consider using the method of cylindrical shells. The formula for the volume using shells is: \(V = 2\pi \int_a^b x \cdot f(x) \, dx\)
Set up the integral for the volume: \(V = 2\pi \int_0^4 x (4 - x)^{-\frac{1}{3}} \, dx\)
Check the behavior of the function near the endpoint \(x=4\) to determine if the integral converges. Since \((4 - x)^{-\frac{1}{3}}\) becomes unbounded as \(x\) approaches 4, analyze the improper integral by considering the limit as \(t \to 4^-\) of the integral from 0 to \(t\).
Evaluate the integral (or determine if it converges) by applying an appropriate substitution, such as \(u = 4 - x\), and then integrate with respect to \(u\). This will help in finding the volume or concluding that it does not exist.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m
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 generated when a region in the plane is revolved around an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the given function.
Recommended video:
04:48
Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around a vertical axis. The volume element is 2π(radius)(height)(thickness). It is especially useful when revolving around the y-axis and when the function is given in terms of x.
Recommended video:
07:33
Euler's Method

Improper Integrals and Convergence

When the region or function involves infinite limits or unbounded behavior, the volume integral may be improper. Determining if the volume exists requires checking the convergence of the integral. If the integral diverges, the volume does not exist or is infinite.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

92–98. Evaluate the following integrals.

94. ∫ (dt / (t³ + 1))

94
views
Textbook Question

Evaluate the following integrals.

∫ x/(x² + 6x + 18) dx

51
views
Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

30. ∫ x³√(1 - x²) dx

80
views
Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

73
views
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

40. ∫[0 to π/6] tan⁵(2x) sec(2x) dx

87
views
Textbook Question

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

69. The region bounded by f(x) = 1/√(x ln x) and the x-axis on the interval [e, ∞) is revolved about the x-axis.

66
views