Shell method versus disk method

In summary, the shell method and the disk method are two techniques used to find the volume of a solid of revolution, with the main difference being the shape of the slices used for calculation. The decision to use one method over the other depends on the shape of the solid and how it is being rotated, with the disk method typically used for circular shapes and the shell method used for rectangular shapes. Neither method is considered easier than the other in all cases, and they both have limitations in terms of the shapes they can be used for and the number of axes of rotation. It is important to use the appropriate method for each situation to avoid incorrect volume calculations.
  • #1
fiziksfun
78
0
I'm confused when to use the shell method vs. the disk method

Specifically in this problem:

If the region encolsed by the y-axis, the line y=2 and the curve y=[tex]\sqrt{x}[/tex] is revolved around the y-axis, the volume of the solid generated is:

However when I use the shell method and the disk method I get different answers

Can someone tell me what I'm doing wrong?

Disk Method:
[tex]\pi[/tex] [tex]\int[/tex]y^4 dy evaluated from 0 to 2

Shell Method:

2[tex]\pi[/tex][tex]\int[/tex]x(2-[tex]\sqrt{x}[/tex])dx

Help please!
 
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  • #2
I got the same answer. Remember the second integral is from 0 to 4 not 0 to 2.
 

What is the difference between the shell method and the disk method?

The shell method and the disk method are two techniques used to find the volume of a solid of revolution, which is a three-dimensional shape formed by rotating a two-dimensional shape around an axis. The main difference between the two methods is the shape of the slices used to calculate the volume.

When should I use the shell method versus the disk method?

The decision to use the shell method or the disk method depends on the shape of the solid of revolution and how it is being rotated. The disk method is typically used when the shape being rotated is a disc or a solid with a circular base. The shell method is used when the shape being rotated is a rectangle or a solid with a rectangular base.

Which method is easier to use?

Both the shell method and the disk method have their own advantages and disadvantages, and neither can be considered easier than the other in all cases. However, the disk method is generally considered easier to use when the shape being rotated is a solid with a circular base, while the shell method is easier to use when the shape has a rectangular base.

What are the limitations of the shell method and the disk method?

The shell method and the disk method have limitations in terms of the shapes they can be used to calculate the volume of. The disk method can only be used for solids with circular bases, while the shell method can only be used for solids with rectangular bases. Additionally, both methods can only be used to find the volume of solids of revolution with a single axis of rotation.

Can the shell method and the disk method be used interchangeably?

No, the shell method and the disk method cannot be used interchangeably. Each method has its own specific conditions and limitations, and using the wrong method for a particular shape can result in an incorrect volume calculation. It is important to understand the differences between the two methods and use the appropriate one for each situation.

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