In this lesson we are going to take a look at using integration to compute volumes by means of the method called Method of Disks and Washers. The Disc Method is used for calculating the volume of a solid of revolution of a solid-state material when integrating along the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of disks of varying radius and infinitesimal thickness. It is also possible to use the same principles with washers instead of disks the Washer Method' to obtain hollow solids of revolutions.
Calculus I - Volumes of Solids of Revolution/Method of Cylinders
We can use this fact as the building block in finding volumes of a variety of shapes. Given an arbitrary solid, we can approximate its volume by cutting it into n thin slices. When the slices are thin, each slice can be approximated well by a general right cylinder. The total volume of the solid is approximately:. Recognize that this is a Riemann Sum. By taking a limit as the thickness of the slices goes to 0 we can find the volume exactly. We recognize this as a definite integral.
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In this section we will start looking at the volume of a solid of revolution. We should first define just what a solid of revolution is. We then rotate this curve about a given axis to get the surface of the solid of revolution. Doing this for the curve above gives the following three dimensional region. What we want to do over the course of the next two sections is to determine the volume of this object.