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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?
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.
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.
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.
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.
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.