There is no general rule for this.Miike012 said:Is there a general rule when to use the shell or washer method when working with calculating volumes of functions rotating about an axis? For instance should I use the shell method when rotating about the y-axis and use the washer method when rotating about the x?
The Shell Method and the Washer Method are two different approaches for calculating the volume of a solid formed by rotating a region around an axis. The main difference between these two methods is the shape of the cross-sections used to calculate the volume. The Shell Method uses cylindrical shells, while the Washer Method uses washers or disks.
The choice between using the Shell Method or the Washer Method depends on the shape of the region being rotated. If the region is bounded by two curves, the Shell Method is typically the easier and more efficient method to use. If the region is bounded by a single curve, the Washer Method is usually the better option. However, it is always a good idea to check if both methods can be applied and choose the one that is simpler to use.
The integral for the Shell Method is set up by integrating the circumference of the cylindrical shells multiplied by the height of the shell. The integral for the Washer Method is set up by integrating the area of the washers or disks multiplied by the thickness of the disk. It is important to correctly identify the bounds of integration and the functions involved in the integrand.
No, the Shell Method and the Washer Method can only be used for regions that can be rotated around an axis. If the region has holes or gaps, a different method, such as the Method of Cylindrical Shells, may need to be used.
One limitation of using the Shell Method and the Washer Method is that they can only be used to calculate volumes of solids of revolution. Additionally, these methods may not be applicable if the region being rotated has a varying density or if the axis of rotation is not perpendicular to the base of the region.