Volumes of Solids of Revolution

# Volumes of Solids of Revolution - Volumes of Solids of...

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Volumes of Solids of Revolution / Method of Rings 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. To get a solid of revolution we start out with a function, , on an interval [ a,b ]. We then rotate this curve about a given axis to get the surface of the solid of revolution. For purposes of this discussion let’s rotate the curve about the x -axis, although it could be any vertical or horizontal axis. 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. In the final the Area and Volume Formulas section of the Extras chapter we derived the following formulas for the volume of this solid.

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where, and is the cross-sectional area of the solid. There are many ways to get the cross-sectional area and we’ll see two (or three depending on how you look at it) over the next two sections. Whether we will use or will depend upon the method and the axis of rotation used for each problem. One of the easier methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation. Doing this the cross section will be either a solid disk if the object is solid (as our above example is) or a ring if we’ve hollowed out a portion of the solid (we will see this eventually). In the case that we get a solid disk the area is, where the radius will depend upon the function and the axis of rotation. In the case that we get a ring the area is, where again both of the radii will depend on the functions given and the axis of rotation. Note as well that in the case of a solid disk we can think of the inner radius as zero and we’ll arrive at the correct formula for a solid disk and so this is a much more general formula to use. Also, in both cases, whether the area is a function of x or a function of y will depend upon the axis of rotation as we will see.
This method is often called the method of disks or the method of rings . Let’s do an example. Example 1 Determine the volume of the solid obtained by rotating the region bounded by , , , and the x -axis about the x -axis. Solution

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Volumes of Solids of Revolution - Volumes of Solids of...

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