Volume - the shell method:
Instead of taking small elements of area such as a rectangle, we need to map out the small element of volume, which is equal to the cross-sectional area multiplied by the interval Δx. Then we will sum them all up to find the total volume of solid. There are a number of ways to come up with this element of volume, but only 2 are really important to this class:
1) The disc method
2) 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:
1. It can be used only if the shape is SYMMETRICAL around some axis.
2. It can be used only if the shape is CIRCULAR around the axis perpendicular to its symmetry.
For example, a cylinder is a great candidate for the shell method. It is symmetrical from the side, and it is circular around axis perpendicular to the side of it. In addition, 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 summarizing them. But, instead of discs, we're defining thin, cylindrical shells.
The cylinder method is used when the slice that was drawn is parallel to axis of the revolution. The method of calculating the volume of a solid of revolution by integrating over volumes of infinitesimal shell-shaped sections bounded by the cylinders with the same axis of revolution as the solid. The best way to compute the volume by this method would be to 1st calculate 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 x-axis.