Using the Shell method to find the volume of a solid

In summary, the Shell method is a mathematical technique used to find the volume of a solid by slicing it into cylindrical shells and integrating their volumes. It is typically used for solids with curved or irregular shapes. To use the Shell method, one must choose an axis of rotation, slice the solid into cylindrical shells, and calculate the volume of each shell using the formula V = 2πrh. It can also be applied to solids with holes or voids. The advantages of using the Shell method include its ability to handle non-uniform cross-sections and its flexibility in integrating both horizontal and vertical slices.
  • #1
Zuni Tiberius
1
0
Use the shell method to find the volume of a solid generated by revolving the region bounded by the given curves and lines about the x-axis.

x=2√y
x=-2y
y=1

So I drew a graph and then using the equation v=∫2πrh

and I got the following

v=∫(from 0 to 2) 2π(y-1)((2√y)-(-2y))

but this is wrong. any suggestions?
 
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  • #2
Zuni Tiberius said:
I got the following

v=∫(from 0 to 2) 2π(y-1)((2√y)-(-2y))

but this is wrong. any suggestions?
Why do you have (y-1) for the radius? And why are your limits 0 to 2?
 

FAQ: Using the Shell method to find the volume of a solid

What is the Shell method for finding the volume of a solid?

The Shell method is a mathematical technique used to find the volume of a solid by slicing it into cylindrical shells and integrating their volumes.

When is the Shell method typically used?

The Shell method is most commonly used when the solid being measured has a curved or irregular shape that cannot be easily measured using other methods such as the Disk or Washer method.

How do you set up the Shell method to find the volume of a solid?

To use the Shell method, you must first choose an axis of rotation and then slice the solid into thin cylindrical shells parallel to that axis. The volume of each shell can then be calculated using the formula V = 2πrh, where r is the radius and h is the height of the shell. Finally, integrate the volumes of all the shells to find the total volume of the solid.

Can the Shell method be applied to solids with holes or voids?

Yes, the Shell method can be applied to solids with holes or voids as long as the holes are circular and parallel to the axis of rotation. The volume of the holes can be subtracted from the total volume of the solid after the integration step.

What are the advantages of using the Shell method?

The Shell method is advantageous because it can be used to find the volume of solids with non-uniform cross-sections, making it a versatile technique for calculating volumes. It also allows for the integration of both horizontal and vertical slices, providing more flexibility in finding the volume of a solid.

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