Many three-dimensional solids can be generated by revolving a curve about the $x$-axis or $y$-axis. For example, if we revolve the semi-circle given by $f(x)=\sqrt{r^2-x^2}$ about the $x$-axis, we obtain a sphere of radius $r$. We can derive the familiar formula for the volume of this sphere.

#### Finding the Volume of a Sphere

Consider a cross-section of the sphere as shown. It is a circle with
radius $f(x)$ and area $\pi [f(x)]^2$. Informally speaking, if we
“slice” the sphere vertically into discs, each disc having
infinitesimal thickness $dx$, the volume of each disc is approximately
$\pi [f(x)]^2\, dx$. If we “add up” the volumes of the discs, we
will get the volume of the sphere:
\begin{eqnarray*}
V&=&\int^r_{-r} \pi [f(x)]^2\, dx\\
&=& \int^r_{-r} \pi (r^2-x^2)\, dx\\
&=& \left.\pi \left(r^2x-\frac{x^3}{3}\right)\right|^r_{-r}\\
&=& \pi \left(\frac{2}{3}r^3\right)-\pi \left(-\frac{2}{3}r^3\right)\\
&=& \frac{4}{3}\pi r^3,\quad{\small\textrm{as expected.}}
\end{eqnarray*}
This is called the **Method of Discs**. In general, suppose
$y=f(x)$ is nonnegative and continuous on $[a,b]$. If the region
bounded above by the graph of $f$, below by the $x$-axis, and on the
sides by $x=a$ and $x=b$ is revolved about the $x$-axis, the volume
$V$ of the generated solid is given by
\[V=\int^a_b \pi [f(x)]^2\, dx.\]
We can also obtain solids by revolving curves about the $y$-axis.

#### Revolving a Region about the $y$-axis

If we revolve the region enclosed by $y=x^2$ and $y=2x$, $0\leq x\leq 2$, about the $y$-axis, we generate a three-dimensional solid.

Let’s find the volume of this solid. If we “slice” the solid horizontally, each slice is a “washer.” The outer radius is $\sqrt{y}$, since $y=x^2 \rightarrow x=\sqrt{y}$, and the inner radius is $y/2$, since $y=2x \rightarrow x=y/2$, and the thickness is $dy$. The volume of each washer is therefore \[ [\pi (\sqrt{y})^2-\pi (y/2)^2]\, dy.\] Then the volume of the entire solid is given by \begin{eqnarray*} \int^4_0 [\pi (\sqrt{y})^2-\pi (y/2)^2]\, dy&=&\int^4_0 \pi \left[y-\frac{y^2}{4}\right]\, dy\\ &=& \left.\pi \left[\frac{y^2}{2}-\frac{y^3}{12}\right]\right|^4_0\\ &=& \pi \left( 8-\frac{16}{3}\right)-\pi \left(0-0\right)\\ &=&\frac{8\pi}{3}. \end{eqnarray*} This generalization of the Method of Discs is called the **Method of Washers**. As we have seen, these methods may be used when a region is revolved about either axis.

We could have taken a different approach in the previous example:

#### Another Method

Look again at the volume of the solid generated by revolving the region enclosed by $y=2x$, $y=x^2$, $0\leq x\leq 2$ about the $y$-axis. This time, we will view the solid as being composed of a collection of concentric cylindrical shells of radius $x$, height $2x-x^2$, and infinitesimal thickness $dx$. The volume of each shell is approximately given by the lateral surface area $2\pi\cdot {\small\textrm{radius}}\cdot {\small\textrm{height}}$ multiplied by the thickness: \[2\pi x[2x-x^2]\, dx.\] “Adding up” the volumes of the cylindrical shells, \begin{eqnarray*} V&=& \int^2_0 2\pi x[2x-x^2]\, dx\\ &=& \int^2_0 2\pi [2x^2-x^3]\, dx\\ &=& \left.\left(\frac{4}{3}\pi x^3-\frac{1}{2}\pi x^4\right)\right|^2_0\\ &=& \left(\frac{32}{3}\pi-8\pi\right)-\left(0-0\right)\\ &=& \frac{8\pi}{3},\quad{\small\textrm{as found earlier.}} \end{eqnarray*} This is called the **Method of Cylindrical Shells**. Suppose $f(x)$, $g(x)$, $F(y)$, $G(y)$ satisfy all the requirements given earlier. Then, for a region revolved about the $y$-axis, \[V=\int_a^b 2\pi xf(x)\, dx \qquad{\small\textrm{or}}\qquad V=\int_a^b 2\pi x[f(x)-g(x)]\, dx.\] For a region revolved about the $x$-axis, \[V=\int_c^d 2\pi yF(y)\, dy \qquad{\small\textrm{or}}\qquad V=\int_c^d 2\pi y[F(y)-G(y)]\, dy.\]

#### Notes

- In the disc and washer methods, you integrate with respect to
the
*same*variable as the axis about which you revolved the region.

- In the method of cylindrical shells, you integrate with respect
to the
*other*variable.

Computing volumes using these methods takes some practice. With experience, you will be better able to visualize the solids and determine which method to apply.

#### Key Concepts

**Method of Washers:**

\begin{eqnarray*} V = \int^b_a \pi ([f(x)]^2 – [g(x)]^2)\, dx & \qquad{\small\textrm{or}}\qquad & V = \int^d_c \pi ([F(y)]^2 – [G(y)]^2)\, dy. \\ \end{eqnarray*}

**Method of Cylindrical Shells:**
\begin{eqnarray*}
V = \int^b_a 2\pi xf(x)\, dx & \qquad{\small\textrm{or}}\qquad &
V = \int^b_a 2\pi x[f(x)-g(x)]\, dx. \\
V = \int^d_c 2\pi yF(y)\, dy & \qquad{\small\textrm{or}}\qquad &
V = \int^d_c 2\pi y[F(y)-G(y)]\, dy. \\
\end{eqnarray*}