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 indefinitely small circular rings. The formula for the volume of any solid of rotation is V=\int\limits_a^b A(x)dx or V=\int\limits_a^b A(y)dy , depending on which axis the rotation is around. In the case of volume by rings, the formula is

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


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, the formula is



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.

&V=\pi\int\limits_a^b f(x)^2dx=\pi\int\limits_0^4(\sqrt{x})^2dx
\\&V=\pi\int\limits_0^4 xdx

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 isolated y . It is important to keep in mind that r is not a variable and must not be treated as one.

\\V&=\pi\int\limits_a^b f(x)^2dx=\pi\int\limits_{-r}^r\big(\sqrt{r^2-x^2}\big)^2dx

See also

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