### Learning Outcomes

- Find the volume of a solid of revolution using the disk method
- Find the volume of a solid of revolution with a cavity using the washer method

## The Disk Method

When we use the slicing method with solids of revolution, it is often called the **disk method** because, for solids of revolution, the slices used to over approximate the volume of the solid are disks. To see this, consider the solid of revolution generated by revolving the region between the graph of the function [latex]f(x)={(x-1)}^{2}+1[/latex] and the [latex]x\text{-axis}[/latex] over the interval [latex]\left[-1,3\right][/latex] around the [latex]x\text{-axis}\text{.}[/latex] The graph of the function and a representative disk are shown in Figure 9(a) and (b). The region of revolution and the resulting solid are shown in Figure 9(c) and (d).

We already used the formal Riemann sum development of the volume formula when we developed the slicing method. We know that

The only difference with the disk method is that we know the formula for the cross-sectional area ahead of time; it is the area of a circle. This gives the following rule.

### The Disk Method

Let [latex]f(x)[/latex] be continuous and nonnegative. Define [latex]R[/latex] as the region bounded above by the graph of [latex]f(x),[/latex] below by the [latex]x\text{-axis,}[/latex] on the left by the line [latex]x=a,[/latex] and on the right by the line [latex]x=b.[/latex] Then, the volume of the solid of revolution formed by revolving [latex]R[/latex] around the [latex]x\text{-axis}[/latex] is given by

The volume of the solid we have been studying (Figure 9) is given by

Let’s look at some examples.

### Example: Using the Disk Method to Find the Volume of a Solid of Revolution 1

Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of [latex]f(x)=\sqrt{x}[/latex] and the [latex]x\text{-axis}[/latex] over the interval [latex]\left[1,4\right][/latex] around the [latex]x\text{-axis}\text{.}[/latex]

### Try It

Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of [latex]f(x)=\sqrt{4-x}[/latex] and the [latex]x\text{-axis}[/latex] over the interval [latex]\left[0,4\right][/latex] around the [latex]x\text{-axis}\text{.}[/latex]

Watch the following video to see the worked solution to the above Try It.

So far, our examples have all concerned regions revolved around the [latex]x\text{-axis,}[/latex] but we can generate a solid of revolution by revolving a plane region around any horizontal or vertical line. In the next example, we look at a solid of revolution that has been generated by revolving a region around the [latex]y\text{-axis}\text{.}[/latex] The mechanics of the disk method are nearly the same as when the [latex]x\text{-axis}[/latex] is the axis of revolution, but we express the function in terms of [latex]y[/latex] and we integrate with respect to [latex]y[/latex] as well. This is summarized in the following rule.

### The Disk Method for Solids of Revolution around the [latex]y[/latex]-axis

Let [latex]g(y)[/latex] be continuous and nonnegative. Define [latex]Q[/latex] as the region bounded on the right by the graph of [latex]g(y),[/latex] on the left by the [latex]y\text{-axis,}[/latex] below by the line [latex]y=c,[/latex] and above by the line [latex]y=d.[/latex] Then, the volume of the solid of revolution formed by revolving [latex]Q[/latex] around the [latex]y\text{-axis}[/latex] is given by

The next example shows how this rule works in practice.

### example: Using the Disk Method to Find the Volume of a Solid of Revolution 2

Let [latex]R[/latex] be the region bounded by the graph of [latex]g(y)=\sqrt{4-y}[/latex] and the [latex]y\text{-axis}[/latex] over the [latex]y\text{-axis}[/latex] interval [latex]\left[0,4\right].[/latex] Use the disk method to find the volume of the solid of revolution generated by rotating [latex]R[/latex] around the [latex]y\text{-axis}\text{.}[/latex]

Figure 11 shows the function and a representative disk that can be used to estimate the volume. Notice that since we are revolving the function around the [latex]y\text{-axis,}[/latex] the disks are horizontal, rather than vertical.

The region to be revolved and the full solid of revolution are depicted in the following figure.

To find the volume, we integrate with respect to [latex]y.[/latex] We obtain

The volume is [latex]8\pi [/latex] units^{3}.