volume of solid of revolutionVolumeOfSolidOfRevolution2007-06-29 11:43:002007-06-30 15:05:01Topicsolid of revolutionvolume of solid% this is the default PlanetMath preamble. as your knowledge
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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 \PMlinkname{intervals}{Interval} \, $[a,\,x_1]$,\, $[x_1,\,x_2]$,\,\ldots,\,$[x_{n-1},\,b]$. The volume $V$ of the solid of revolution thus satisfies
$$V_< \le V \le V_>,$$
and this is true for any \PMlinkescapetext{division}\, $x_1 < x_2 < \ldots < x_{n-1}$\, 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
$$V = \pi\int_a^b[f(x)]^2\,dx.$$
\begin{thebibliography}{8}
\bibitem{Lj}{\sc E. Lindel\"of}: {\em Johdatus korkeampaan analyysiin}. Nelj\"as painos.\, Werner S\"oderstr\"om Osakeyhti\"o, Porvoo ja Helsinki (1956).
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