Volume by rings

Volume by rings, also known as volume by disks or volume by washers (if the area between two functions is being rotated around an axis), is a method of finding the volume of a solid of revolution. This method involves splitting the shape into infinitely small circular rings and summing them up. The formula for the volume of any solid of rotation is $V=\int\limits_a^b A(x)dx$, where $A(x)$ denotes an area function.In the case of volume by rings, the formula is

$V=\pi\int\limits_a^b f(x)^2dx$

assuming the rotation is around the x-axis. If the rotation is of an area between two functions $f(x)$ and $g(x)$, the formula is

$V=\pi\int\limits_a^b\bigl(f(x)^2-g(x)^2\bigr)dx$

Examples

To find the volume of the resulting solid when $f(x)=\sqrt{x}$ is rotated around the $x$-axis on the interval $(0,4)$, substitute into the formula.

$\begin{align} &V=\pi\int\limits_a^b f(x)^2dx=\pi\int\limits_0^4(\sqrt{x})^2dx \\&V=\pi\int\limits_0^4 xdx \\&V=\pi\left[\frac{x^2}{2}\right]_0^4=\pi\left(\frac{4^2}{2}-\frac{0^2}{2}\right)=\pi(8-0)=8\pi\end{align}$

This method can also be used to find the formula for the volume of shapes. Take for example the formula for the volume of a sphere, $V=\frac{4\pi}{3}r^3$. A sphere is a graph of $x^2 + y^2 = r^2$ rotated around an axis (here we will assume it is the $x$-axis). Begin by isolating $y$. It is important to keep in mind that $r$ is not a variable and must not be treated as one.

$\begin{align}y&=\sqrt{r^2-x^2} \\V&=\pi\int\limits_a^b f(x)^2dx=\pi\int\limits_{-r}^r\big(\sqrt{r^2-x^2}\big)^2dx \\&=\pi\int\limits_{-r}^r(r^2-x^2)dx \\&=\pi\left[r^2x-\frac{x^3}{3}\right]_{-r}^r \\&=\pi\left(\left(r^2(r)-\frac{r^3}{3}\right)-\left(r^2(-r)-\frac{(-r)^3}{3}\right)\right)=\pi\left(\left(r^3-\frac{r^3}{3}\right)-\left(-r^3+\frac{r^3}{3}\right)\right) \\&=\pi\left(2r^3-\frac{2r^3}{3}\right)=\pi\left(\frac{4r^3}{3}\right)=\frac{4\pi}{3}r^3\end{align}$

See also

• Volume by shells