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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 $, 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

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