The volume of a solid of revolution^{} (http://planetmath.org/VolumeOfSolidOfRevolution) can be obtained from

$$V={\int}_{a}^{b}\pi {[f(x)]}^{2}\mathit{d}x,$$ 

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 (http://planetmath.org/AngleBetweenTwoLines) $\omega $ with the normal line of those planes, then we have the volume formula of the form

$$V={\int}_{a}^{b}A(t)\mathit{d}t\mathrm{cos}\omega .$$ 
