Volume by disks, shells, or washers?

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SUMMARY

The discussion focuses on determining the appropriate method for calculating the volume of solids of revolution using disks, shells, or washers. The consensus is that the shell method is preferred for complex shapes, while the disk method is suitable for simple shapes without holes. The washer method is specifically used when a hole exists in the solid. Ultimately, the choice of method depends on the functions that define the region being revolved and the complexity of the shape involved.

PREREQUISITES
  • Understanding of solid geometry and volume calculation
  • Familiarity with methods of integration in calculus
  • Knowledge of the disk, shell, and washer methods for volume calculation
  • Ability to identify functions that bound regions in the Cartesian plane
NEXT STEPS
  • Study the shell method for volume calculation in detail
  • Practice problems involving the disk and washer methods
  • Explore integration techniques for evaluating complex integrals
  • Learn about the applications of volume calculation in real-world scenarios
USEFUL FOR

Students and educators in calculus, mathematicians, and anyone involved in solid geometry or volume calculations will benefit from this discussion.

KMcFadden
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What is the easiest way to determine if the volume of a solid should be found by using disks, shells, or washers?
 
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Personally I always start with the shell method unless it is a really basic, simple shape(disk method)... But to answer ur question, I look for a hole. If there is one, then i am thinking washer, if there isn't and the solid is a very simple shape, then I go with the disk method. For the most complex shapes, i automatically go to the shell method.
 
KMcFadden said:
What is the easiest way to determine if the volume of a solid should be found by using disks, shells, or washers?
There is no one-size-fits-all guidance here. It depends on the function/functions that bound the region being revolved. The disk and washer methods are essentially the same, with the washer method being used if the typical volume element is a disk with a hole in it.

Sometimes one method produces an integral that's easier to evaluate, and sometimes another method does.
 

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