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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

or

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

V=\pi\int\limits_a^b\big(f(x)^2-g(x)^2\big)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 isolated 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
\\V&=\pi\int\limits_{-r}^r(r^2-x^2)dx
\\V&=\pi\left[r^2x-\frac{x^3}{3}\right]_{-r}^r
\\V&=\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)
\\V&=\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

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