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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 6.4.63a
Chapter 6, Problem 6.4.63a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution.

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Recall the shell method is used to find volumes of solids of revolution by integrating cylindrical shells formed by revolving a region around an axis.
Understand that in the shell method, each shell is generated by revolving a vertical or horizontal strip around the axis of revolution.
Note that the axis of each cylindrical shell is actually perpendicular to the axis of revolution, because the shell is formed by rotating a line segment around that axis.
Therefore, the statement 'the axis of the cylindrical shells is parallel to the axis of revolution' is false.
A counterexample: If the axis of revolution is the x-axis, the shells are vertical cylinders whose axes are parallel to the y-axis, which is perpendicular to the x-axis.

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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 Calculation

The shell method is a technique for finding the volume of a solid of revolution by integrating cylindrical shells. It involves slicing the solid into thin shells parallel to the axis of revolution and summing their volumes. Understanding how these shells are formed and their dimensions is crucial to applying the method correctly.
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Finding Volume Using Disks

Axis of Revolution

The axis of revolution is the line about which a region is rotated to generate a solid. It serves as a reference for determining the shape and orientation of the solid and the shells or disks used in volume calculations. Recognizing the axis helps in setting up the integral and choosing the appropriate method.
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Disk Method Using y-Axis

Orientation of Cylindrical Shells Relative to the Axis

In the shell method, cylindrical shells are formed by revolving vertical or horizontal slices around the axis of revolution. Importantly, the axis of each shell is perpendicular, not parallel, to the axis of revolution. This orientation affects how the radius and height of each shell are defined in the integral.
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Related Practice
Textbook Question

Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

a. On what intervals is the object moving in the positive direction?

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.

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Textbook Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


a. Graph the velocity function for both riders. 

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Textbook Question

Calculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position

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Textbook Question

21–30. {Use of Tech} Arc length by calculator


a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = ln x, for 1≤x≤4

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Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

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