
We use the procedure of “Slice, Approximate, Integrate” to develop the washer method to compute volumes of solids of revolution.

### The washer method

We can slice a solid of revolution perpendicular to the axis of rotation. We saw that we could generate the solid of revolution by considering the corresponding slices in the region of revolution in the $xy$-plane. To illustrate the details, we start with a motivating example.

### The washer method formula

Let’s generalize the ideas in the above example. First, note that we slice the region of revolution perpendicular to the axis of revolution, and we approximate each slice by a rectangle. We call the slice obtained this way a washer. If the washer is not hollow (i.e. $r=0$), it is sometimes referred to as a disk. Washers are characterized by finite inner and outer radii but infinitesimal height. We now summarize the results of the above argument.

At the risk of belaboring an important point, the variable of integration is determined by the requirement that the slices be perpendicular to the axis of rotation. Indeed, this requirement will allow us to determine if the slices are vertical or horizontal. Just as in the previous applications, if we have vertical slices, we will integrate with respect to $x$, and if we have horizontal slices, we will integrate with respect to $y$.

Now, let’s see how this formula works in action by considering an example where we take the same region and revolve it about different lines.