Skip to main content
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 6.4.63b
Chapter 6, Problem 6.4.63b

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. If a region is revolved about the y-axis, then the shell method must be used.

Verified step by step guidance
1
Understand the problem: The statement claims that if a region is revolved about the y-axis, then the shell method must be used. We need to determine if this is true or false and explain why.
Recall the two common methods for finding volumes of solids of revolution: the disk/washer method and the shell method. Both methods can be used depending on the axis of revolution and the shape of the region.
Analyze the disk/washer method: This method involves slicing the solid perpendicular to the axis of revolution. When revolving around the y-axis, slices perpendicular to the y-axis are horizontal slices, which can be used with the disk/washer method if the function is expressed as x in terms of y.
Analyze the shell method: This method involves slicing the solid parallel to the axis of revolution. When revolving around the y-axis, vertical slices are used to form cylindrical shells, which is often convenient if the function is expressed as y in terms of x.
Conclusion: The shell method is not the only method that can be used when revolving around the y-axis. The disk/washer method can also be used if the problem is set up appropriately. Therefore, the statement is false because the shell method is not mandatory.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Shell Method

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells formed by revolving vertical or horizontal slices around an axis. It is especially useful when the axis of rotation is parallel to the slices, often simplifying the integral setup.
Recommended video:
07:33
Euler's Method

Disk/Washer Method

The disk or washer method finds volumes by slicing the solid perpendicular to the axis of rotation, creating circular cross-sections. It is typically used when the region is revolved around an axis and the slices are easy to express as functions of the variable of integration.
Recommended video:
06:30
Disk Method Using y-Axis

Choice of Method Depends on the Axis and Region

The method used to find volume depends on the axis of rotation and the shape of the region. Revolving around the y-axis does not mandate the shell method; sometimes the disk/washer method is simpler or possible, depending on how the region is described and the variable of integration.
Recommended video:
06:30
Disk Method Using y-Axis
Related Practice
Textbook Question

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.


b. What is the height of a cylindrical shell at a point x in [0, 4]?

Textbook Question

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


b. Find the area of region R₂ using geometry and the answer to part (a).


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.

Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


b. Find the displacement over the given interval. 


v(t) = 3t²−6t on [0, 3]

Textbook Question

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.


b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

1
views
Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.


b. Find the function that gives the total blood pumped between t=0 and a future time t>0.

Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


b. Suppose P=10, A=20, and r=0. If the initial population is N(0)=100, does the population ever become extinct? Explain.