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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 6.4.34
Chapter 6, Problem 6.4.34

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. 


{Use of Tech} y² = ln x,y² = ln x³, and y=2; about the x-axis

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First, identify the region R bounded by the curves given: \(y^2 = \ln x\), \(y^2 = \ln x^3\), and \(y = 2\). Rewrite the second curve using logarithm properties: \(y^2 = \ln x^3 = 3 \ln x\).
Since the region is revolved about the x-axis, and the shell method involves cylindrical shells formed by revolving vertical slices, determine the variable of integration. Here, it is easier to integrate with respect to \(y\) because the axis of rotation is horizontal (x-axis).
Express \(x\) in terms of \(y\) from the given curves: from \(y^2 = \ln x\), we get \(x = e^{y^2}\); from \(y^2 = 3 \ln x\), we get \(x = e^{y^2/3}\). These represent the left and right boundaries of the region for each fixed \(y\).
Set up the shell radius and height for the shell method. The radius of a shell is the distance from the shell to the axis of rotation, which is \(y\) (since we revolve around the x-axis). The height of the shell is the horizontal length between the two curves, which is \(e^{y^2/3} - e^{y^2}\).
Determine the limits of integration for \(y\). Since \(y\) is bounded above by \(y=2\) and below by the intersection of the curves or \(y=0\) (depending on the region), integrate the volume using the shell method formula: \(V = 2\pi \int_{y=0}^{2} (\text{radius})(\text{height}) \, dy = 2\pi \int_0^2 y \left(e^{\frac{y^2}{3}} - e^{y^2}\right) dy\).

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Key Concepts

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

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell's volume is approximated by its circumference times height times thickness. This method is especially useful when the axis of rotation is parallel to the axis of the variable of integration.
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Finding Volume Using Disks

Understanding the Region Bounded by Curves

Identifying the region R involves analyzing the given curves y² = ln x, y² = ln x³, and y = 2. Understanding their intersections and boundaries is crucial to set correct integration limits and expressions for shell height and radius.
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Finding Area When Bounds Are Not Given

Revolution About the x-axis

Revolving the region about the x-axis means the shells are formed perpendicular to this axis. This affects how the radius and height of each shell are expressed in terms of y or x, guiding the setup of the integral for volume calculation.
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Disk Method Using y-Axis
Related Practice
Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.


y=√sin x,y=1, and x=0; about the x-axis 

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Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.


y=3x+4, for 0≤x≤6; about the x-axis

Textbook Question

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.


v(t) = t(25−t²)^1/2, for 0≤t≤5

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Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

Textbook Question

The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.

Textbook Question

Find the volume of the torus formed when the circle of radius 2 centered at (3, 0) is revolved about the y-axis. Use geometry to evaluate the integral.

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