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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.4.16
Chapter 6, Problem 6.4.16

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.
Graph showing a region bounded by curves, illustrating the shell method for volume calculation when revolved about the x-axis.
x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

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Step 1: Understand the problem. The region R is bounded by the curves x = 4 / (y + y³), x = 1/√3, and y = 1. The region is revolved about the x-axis, and we are tasked with finding the volume of the solid using the shell method.
Step 2: Recall the shell method formula for volume. The volume of a solid of revolution using the shell method is given by: V = ∫[a to b] 2π(radius)(height) dy, where 'radius' is the distance from the axis of rotation (x-axis in this case) and 'height' is the length of the shell (difference between the x-values of the curves).
Step 3: Identify the radius and height. The radius is the distance from the x-axis, which is simply y. The height is the difference between the x-values of the curves: height = (4 / (y + y³)) - (1/√3).
Step 4: Set up the integral. The bounds for y are from y = 1 to y = √3 (as indicated by the graph). The integral becomes: V = ∫[1 to √3] 2π(y)((4 / (y + y³)) - (1/√3)) dy.
Step 5: Simplify and prepare for evaluation. Factor out constants where possible, and simplify the integrand. The integral can then be evaluated using standard techniques of integration, such as substitution or partial fractions, depending on the complexity of the integrand.

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Key Concepts

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

Shell Method

The shell method is a technique for finding the volume of a solid of revolution. It involves integrating the lateral surface area of cylindrical shells formed by revolving a region around an axis. The formula for volume using the shell method is V = 2π ∫ (radius)(height) dy or dx, depending on the axis of rotation.
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Volume of Revolution

The volume of revolution refers to the volume of a three-dimensional shape created by rotating a two-dimensional area around an axis. This concept is fundamental in calculus, as it allows for the calculation of volumes using integration techniques, such as the disk, washer, and shell methods, depending on the orientation of the rotation.
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Bounded Region

A bounded region in calculus is an area enclosed by curves or lines on a graph. In this problem, the region R is defined by the curves x = 4/(y + y³), x = 1/√3, and y = 1. Understanding the boundaries of the region is crucial for setting up the integral correctly when applying the shell method to find the volume.
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Related Practice
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