# volume of solid of revolution

Let us consider a solid of revolution, which is generated when a planar domain $D$ rotates about a line of the same plane. We chose this line for the $x$-axis, and for simplicity we assume that the boundaries of $D$ are the mentioned axis, two ordinates$x=a$,  $x=b\,(>a)$, and a continuous curve   $y=f(x)$.

Between the bounds $a$ anb $b$ we fit a sequence of points  $x_{1},\,x_{2},\,\ldots,\,x_{n-1}$  and draw through these the ordinates which divide the domain $D$ in $n$ parts. Moreover we form for every part the (maximal) inscribed and the (minimal) circumscribed rectangle. In the revolution of $D$, each rectangle generates a circular cylinder. The considered solid of revolution is part of the volume $V_{>}$ of the union of the cyliders generated by the circumscribed rectangles and at the same time contains the volume $V_{<}$ of the union of the cylinders generated by the inscribed rectangles.

Now it is apparent that

 $V_{>}=\pi[M_{1}^{2}(x_{1}-a)+M_{2}^{2}(x_{2}-x_{1})+\ldots+M_{n}^{2}(b-x_{n-1}% )],$
 $V_{<}=\pi[m_{1}^{2}(x_{1}-a)+m_{2}^{2}(x_{2}-x_{1})+\ldots+m_{n}^{2}(b-x_{n-1}% )],$

where  $M_{1},\,M_{2},\,\ldots,\,M_{n}$  are the greatest and  $m_{1},\,m_{2},\,\ldots,\,m_{n}$  the least values of the continuous function $f$ on the intervals (http://planetmath.org/Interval)   $[a,\,x_{1}]$,  $[x_{1},\,x_{2}]$, …, $[x_{n-1},\,b]$. The volume $V$ of the solid of revolution thus satisfies

 $V_{<}\leq V\leq V_{>},$

and this is true for any   $x_{1}  of the interval  $[a,\,b]$. The theory of the Riemann integral guarantees that there exists only one real number $V$ having this property and that it is also the definition of the integral $\displaystyle\int_{a}^{b}\!\pi[f(x)]^{2}\,dx.$ Therefore the volume of the given solid of revolution can be obtained from

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

## References

• 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title volume of solid of revolution VolumeOfSolidOfRevolution 2013-03-22 17:20:12 2013-03-22 17:20:12 pahio (2872) pahio (2872) 11 pahio (2872) Topic msc 51M25 PappussTheoremForSurfacesOfRevolution SurfaceOfRevolution VolumeAsIntegral