Integration Volume: Disc vs Shell - What's the Difference?

In summary, there are two types of volume integration: disk integration and shell integration. The main difference between them is the axis of rotation, with disk integration using a parallel axis and shell integration using a perpendicular axis. To find the general vector equation of a solid of revolution, disks use the radius as the function value and shells use the x-value. This can be calculated by multiplying the circumference (2πx) by the height of the cylinder (f(x)) and integrating. Further clarification may be needed, but this is the basic difference between disk and shell integration.
  • #1
pavadrin
156
0
there are the two types of volume integration which i am aware of, disc-integration and shell integration. What is the difference between these? where would each one? Also i was looking at shell integration on carious sites on the net, and i am still a little confused how the generic formula works. Could someone ever so kindly explain this to me?

many thanks,
pavadrin
 
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  • #2
The difference between disk and shell integration is where the axis of rotation is. Let's assume that you are integrating a solid of rotation of some f(x)dx. If the axis of rotation is parallel to the x-axis then you use disk integration, but if the axis of rotation is perpendicular to the x-axis then you use shell integration.
 
  • #3
How would you find the general vector equation of a solid of revolution?
 
  • #4
"Disks" works, of course, by using disks- normally, the radius of the disk is the value of the function and you calculate the area by [itex]\pi f(x)[/itex], multiply by the "thickness", dx, and then integrate.

"Shells" works by using thin cylinders. The radius is typically the x-value so you have a "circumference" calculation [itex]2\pi x[/itex] and then multiply by the "height" of the cylinder, f(x): you integrate [itex]2\pi x f(x) dx[/itex\.
 
  • #5
okay thanks for the replies
it is a little less confusing now
ill try reading into in more and repost if i am still stuck,
thanks once again for your time,
pavadrin
 

FAQ: Integration Volume: Disc vs Shell - What's the Difference?

What is the difference between disc and shell integration volume methods?

The disc method and shell method are two different techniques used to find the volume of a solid of revolution. The disc method involves slicing the solid into discs perpendicular to the axis of revolution, while the shell method involves slicing the solid into cylindrical shells parallel to the axis of revolution.

When should I use the disc method and when should I use the shell method?

The choice between disc and shell method depends on the shape of the solid of revolution. The disc method is typically used when the cross-sections of the solid are circles, while the shell method is used when the cross-sections are rectangles or squares.

How do I set up the integrals for disc and shell methods?

The setup for both disc and shell integrals involves finding the radius, height, and thickness of each disc or shell. For the disc method, the radius is the distance from the axis of revolution to the disc, and the height is the thickness of the disc. For the shell method, the radius is the distance from the axis of revolution to the center of the shell, and the height is the height of the shell. The thickness is the same for both methods and represents the change in the variable of integration.

Can I use both disc and shell methods to find the volume of the same solid?

Yes, it is possible to use both disc and shell methods to find the volume of the same solid. However, the choice of which method to use may affect the complexity of the integral and the amount of work required to solve it.

Are there any limitations to using disc and shell methods for integration volume?

Both disc and shell methods have limitations, especially when the cross-sections of the solid are not simple shapes. In those cases, other integration methods such as the washer method or the method of cylindrical shells may be more appropriate.

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