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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 $V=\int\limits_a^b A(x)dx$ 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

\begin{align}V&=\int\limits_a^b 2\pi x\cdot f(x)dx \\V&=2\pi\int\limits_a^b x\cdot f(x)dx\end{align}

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 , 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 . 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}