
Many threedimensional solids can be generated by revolving a curve about the $x$axis or $y$axis. For example, if we revolve the semicircle given by $f(x)=\sqrt{r^2x^2}$ 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 SphereConsider a crosssection 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^2x^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 $y$axisIf we revolve the region enclosed by $y=x^2$ and $y=2x$, $0\leq x\leq 2$, about the $y$axis, we generate a threedimensional 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 $y=x^2 \rightarrow x=\sqrt{y}$), 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(00\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 MethodLook 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 $2xx^2$, 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: \[2\pi x[2xx^2]\, dx.\] "Adding up" the volumes of the cylindrical shells, \begin{eqnarray*} V&=& \int^2_0 2\pi x[2xx^2]\, dx\\ &=& \int^2_0 2\pi [2x^2x^3]\, dx\\ &=& \left.\left(\frac{4}{3}\pi x^3\frac{1}{2}\pi x^4\right)\right^2_0\\ &=& \left(\frac{32}{3}\pi8\pi\right)\left(00\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, \[V=\int_c^d 2\pi yF(y)\, dy \qquad{\small\textrm{or}}\qquad V=\int_c^d 2\pi y[F(y)G(y)]\, dy.\]
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 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*} 