Shell method VS washer/disk method

In summary, the choice between using the shell method or the washer/disc method for finding the volume of a rotating object depends on the ease of writing y as a function of x or x as a function of y and the orientation of the axis of rotation. If y is a function of x and the axis is parallel to the x-axis, the disk method is more suitable. On the other hand, if y is a function of x and the axis is parallel to the y-axis, the shell method should be used. The same applies for x as a function of y. It is recommended to choose the method that suits the problem best.
  • #1
Pixleateit
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i'm confused right now about this, when would it be appropriate to use the shell method rather than using the washer or disk method when it comes to looking for the volume of a rotating object?
 
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  • #2
Often either can be used. It really depends upon whether it is easier to write y as a function of x or x as a function of y and whether the axis of rotation is parallel to the x or y axis.

If you have y as a function of x, and the axis of rotation is parallel to the x-axis, then your "disk radius" will depend on y but you will be integrating with respect to x so you will, as you want, be integrating a function of x with respect to x. That would be using the disk method.

If you have y as a function of x, and the axis of rotation is parallel to the y-axis, trying to use the disk method, your "disk radius" will depend on x but you want to integrate with respect to y and x is NOT a function of y. That will cause difficulties. Use the shell method here.

If you have x as a function of y, reverse the two above. Of course, if you can easily write both y as a function of x and x as a function of y, it doesn't matter.

This makes much more sense for a specific problem than as a general rule!
 
  • #3
washers/discs are more suitable in terms of x,y,z coords, roughly, and shells in terms of polar coords, so yes i would say to use the latter on a problem with rotational symmetry.

i.e. they are both the same method just expressed in different coords. so use whichever one suits the problem, as halls suggests.
 

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

The shell method and washer/disk method are both techniques used in calculus to find the volume of a solid of revolution. The main difference between the two methods is how they slice the solid. In the shell method, the solid is sliced vertically into thin cylindrical shells, whereas in the washer/disk method, the solid is sliced horizontally into thin circular disks or washers.

Which method should I use to find the volume of a solid of revolution?

The method you should use depends on the shape of the solid and the axis of rotation. If the solid is rotated around a vertical axis, the shell method is more suitable. If the solid is rotated around a horizontal axis, the washer/disk method is more appropriate. It is also important to consider which method will result in a simpler integration process.

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

Yes, you can use both methods to find the volume of the same solid, but the results may differ. This is because the two methods use different cross-sectional areas to calculate the volume. However, if the solid has a symmetrical shape, both methods should result in the same volume.

Which method is easier to use?

The ease of use of each method depends on the specific problem at hand. In some cases, the shell method may result in simpler integrals, while in other cases, the washer/disk method may be easier. It is recommended to practice both methods and determine which one works best for you in different situations.

Are there any other methods to find the volume of a solid of revolution?

Yes, there are other methods such as the cylindrical shells method and the Pappus's centroid theorem. However, the shell method and washer/disk method are the most commonly used in calculus courses. It is important to understand these two methods before moving on to more complex techniques.

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