What is a solid of revolution?

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We define a solid of revolution and discuss how to find the volume of one in two different ways.

Solids of revolution

Given a region $R$ in the $xy$ -plane, we built solids by stacking “slabs” with given cross sections on top of $R$ . Another way to generate a solid from the region $R$ is to revolve it about a vertical or horizontal axis of revolution. A solid generated this way is often called a solid of revolution. We will be interested in computing the volume of such solids. Before we dive into the details, we first study two ways we can generate such a solid.

Consider a cone with a height of $4$ units and a base radius of $2$ units. We can superimpose the cone on a coordinate system, as shown below.

We can imagine this as a solid of revolution as follows. First, note that the line segment in Quadrant I of the $xy$ -plane that joins the points $(0,4)$ and $(2,0)$ has slope $\answer [given]{-2}$ . Using the point-slope form for a line and the point $(2,0)$ gives the equation of this line segment, which holds for $0 \leq x \leq 2$ .

$\begin{align*} y-y_0 &= m(x-x_0) \\ y - 0 & = -2(x-2) \\ 2x+y &= 4 \end{align*}$

Now, consider the region $R$ in the $xy$ -plane bounded by $2x+y=4$ , $y=0$ , and $x=0$ .

Note that when we revolve $R$ about the , we obtain the cone in question. Since revolving this region about the axis of revolution produced the solid of revolution, we call $R$ in the $xy$ -plane the region of revolution.

There are two ways that we can imagine the solid.

Slicing Perpendicular to the axis of revolution

We can slice the solid perpendicular to the axis of revolution. The axis of revolution is the $y$ -axis, so perpendicular slices are horizontal. A representative slice is shown on the solid below.

The region in the $xy$ -plane that generates the slice once rotated about the $y$ -axis is shown as well.

Ultimately, we need to approximate the slices by a shape whose volume we can compute. Note that the slice on the solid here can be approximated by a disk.

Note that if we approximate the slice in the region of revolution by a rectangle, the disk on the solid is generated by revolving this rectangle about the axis of revolution.

Since we can find the volume of a disk, this seems like a fruitful approach. Before continuing, there are a few important points to summarize.

We slice the solid into many pieces, and we can view each slice as being generated by revolving a certain region in the $xy$ -plane about the axis of revolution.
We can approximate each slice on the solid by a disk, and we can generate this disk by approximating the corresponding slice in the $xy$ -plane by a rectangle.
Thus, we can work with the solid of revolution by analyzing the slices in the $xy$ -plane.

Note here that since one of the sides of the solid lies on the axis of revolution, we obtain a solid disk, but if the axis of revolution does not coincide with the boundary of the region, we instead obtain a disk with a hole in it.

We’ll explore the details in a bit, but let’s first explore another way to generate the solid.

Slicing Parallel to the axis of revolution

We instead can choose to slice the region of revolution in the $xy$ -plane by using slices that are parallel to the axis of revolution. we can approximate each of those slices by a rectangle.

Revolving these about the axis of revolution gives very thin, hollow cylinders.

Now, the solid is built by nesting these hollow cylinders inside of each other, whereas before, when the slices were taken perpendicular to the axis of revolution, we built the solid by stacking the disks on top of each other.

Volumes of Hollow Cylinders

We have seen that we can slice the region of revolution perpendicular or parallel to the axis of revolution. We can then approximate each slice by a rectangle and revolve them about the axis of revolution. In each case, we obtain a (potentially) hollow cylinder. Let’s turn our attention to studying the volume of a hollow cylinder with outer radius $R$ , inner radius $r$ , and height $h$ . A picture is shown below.

To find the volume of the hollow cylinder, note that:

$\begin{align*} \left< \textrm{Volume of the hollow cylinder} \right> &= \\ \left< \textrm{ Volume of the outer } \right. & \left. \textrm{cylinder } \right> - \left< \textrm{ Volume of the inner cylinder} \right>. \end{align*}$

The volume of the outer cylinder, which has radius $R$ is $V_{outer} = \pi R^2 h$ , while the volume of the inner cylinder, which has radius $r$ is $V_{inner} = \pi r^2 h$ .

$\left < \textrm {Volume of the hollow cylinder} \right > = \pi R^2h-\pi r^2h=\pi (R^2-r^2)h$

Note that this requires only three geometric quantities of interest - an outer radius, and inner radius, and a thickness. It does not require that we work with a coordinate system. To use calculus, however, we must work with functions described using a coordinate system. We thus will continue to imagine solids of revolution by revolving regions in the $xy$ -plane about an axis of revolution. One of the essential skills to find the resulting volumes will be a familiar one; we must express $R$ , $r$ , and $h$ in terms of our variable of integration. We explore this in detail in the coming sections.