Volume by shells

Volume by shells is a method of finding the volume of a solid of revolution. This method involves splitting the shape into indefinitely small rectangles folded into a cylinder shape. The formula for the volume of any solid of rotation is $V=\int\limits_a^b A(x)dx$, where $A(x)$ is an area function. This can be applied to any axis of rotation. In the case of volume by rings, the formula is

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

assuming the rotation is around the $y$-axis.

Examples

To find the volume of the resulting solid when $y=\sqrt x$ is rotated around the $x$-axis on the interval $0<x<4$, one must first solve the equation for $x$. This gives $x=y^2$. The interval must then be changed into an interval of $y$, giving $0<y<2$. Now it can be substituted into the formula.

$\begin{align}V&=2\pi\int\limits_a^b y\cdot f(y)dy=2\pi\int\limits_0^2(y^2y)dy \\V&=2\pi\int\limits_0^2 y^3dy \\V&=2\pi\left[\frac{y^4}{4}\right]_0^2=2\pi\left(\frac{2^4}{4}-\frac{0^4}{4}\right)=2\pi(4-0)=8\pi\end{align}$

See also

• Volume by rings