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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 6.4.67
Chapter 6, Problem 6.4.67

64–68. Shell method Use the shell method to find the volume of the following solids.


The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9

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Identify the solid: We have a right circular cone with radius 6 and height 9, and a cylindrical hole of radius 3 drilled along its central axis. The goal is to find the volume of the remaining solid using the shell method.
Set up the coordinate system: Place the cone so that its vertex is at the origin (0,0) and its base is at height 9 along the y-axis. The radius of the cone at height y is proportional to y, given by the linear relation \(r(y) = \frac{6}{9}y = \frac{2}{3}y\).
Express the volume of a typical cylindrical shell: At height y, the shell has radius \(r_{shell} = y\) (distance from the axis), height equal to the thickness \(dy\), and the shell's length is the circumference of the shell times the thickness. The shell's thickness is \(dy\), and the shell's height is the difference between the outer radius and the hole radius, i.e., \(\text{height} = r(y) - 3\).
Write the volume element for the shell: The volume of a thin cylindrical shell is given by \(dV = 2\pi \times (\text{radius of shell}) \times (\text{height of shell}) \times (\text{thickness})\). Here, \(dV = 2\pi y (r(y) - 3) dy\).
Set the limits of integration and write the integral: Since the hole radius is 3, the shell method applies for \(y\) values where \(r(y) \geq 3\). Find the corresponding \(y\) values and integrate \(dV\) from the lower to upper limit to find the volume of the remaining solid.

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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, and integrating these shells over the given interval yields the total volume.
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Geometry of a Right Circular Cone

A right circular cone has a circular base and a vertex aligned perpendicularly above the base center. Its radius and height define its shape, and understanding the linear relationship between radius and height is essential for setting up integrals involving cross-sectional areas.
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Volume of a Solid with a Cylindrical Hole

When a cylindrical hole is drilled through a solid, the volume of the hole must be subtracted from the original volume. Using the shell method, the volume of the remaining solid can be found by integrating the difference between the outer radius and the hole radius along the axis.
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Related Practice
Textbook Question

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.

y = √x,y=0, and x=4; about the x-axis

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Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x² and y = 2−x²; about the x-axis

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Textbook Question

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. 


y = x³−x⁸+1,y=1; about the y-axis

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Textbook Question

Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).

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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=ln x,y=ln x^2; and y=ln 8; about the y-axis

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Textbook Question

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)

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