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# volume as integral

The volume of a solid of revolution can be obtained from

$V\;=\;\int_{a}^{b}\pi[f(x)]^{2}\,dx,$ |

where the integrand is the area of the intersection disc of the solid of revolution and a plane perpendicular to the axis of revolution at a certain value of $x$. This volume formula may be generalized to an analogous formula containing instead of the area $\pi[f(x)]^{2}$ a more general intersection area $A(t)$ obtained from a given solid by cutting it with a set of parallel planes determined by the parameter $t$ on a certain axis. One must assume that the function $t\mapsto A(t)$ is continuous on an interval $[a,\,b]$ where $a$ and $b$ correspond to the “ends” of the solid. If the $t$-axis forms an angle $\omega$ with the normal line of those planes, then we have the volume formula of the form

$V\;=\;\int_{a}^{b}\!A(t)\,dt\,\cos\omega.$ |

## Mathematics Subject Classification

51M25*no label found*51-00

*no label found*

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