Volume:The Shell Method

Instead of taking small elements of area like a rectangle, we need to map out a small element of volume, which is equal to the cross-sectional area multiplied by our interval Δx. Then we'll sum them all up to find the total volume of the solid. There are several ways to come up with this element of volume, but only two are really important to this class:

• The disc method
• The shell method

The "Shell" method

The shell method is a method that can be used when the disc method becomes difficult to use, such as with very complex shapes.

However, there are tighter restrictions on when you can use the Shell method:

• It can only be used if the shape is SYMMETRICAL around some axis.
• It can only be used if the shape is CIRCULAR around the axis perpendicular to its symmetry.

For instance, a cylinder is a great candidate for the shell method. It is symmetrical from the side, and it is circular around the axis perpendicular to its side (the top). Additionally, volumes of revolution always fulfill these requirements.

The shell method is similar to the disc method in that we're defining an element of volume and then summing them up.(an integral). But, instead of discs, we're defining thin, cylindrical shells.

To find the volume of a solid of revolution with the shell method, use on of the following.
Horizontal axis of revolution; Vertical axis of revolution

The cylinder method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution. A method of computing the volume of a solid of revolution by integrating over the volumes of infinitesimal shell-shaped sections bounded by cylinders with the same axis of revolution as the solid. The best way to compute the volume by this method would be to first compute the volume corresponding to the region bounded by y1 and the x-axis, and then to subtract out the volume of rotation of the region between y2 and the x-axis.