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 $[a,\,x_1]$ $[x_1,\,x_2]$ ...,$[x_{n-1},\,b]$ The volume $V$ of the solid of revolution thus satisfies $$V_< \le V \le V_>,$$ and this is true for any 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 $$V = \pi\int_a^b[f(x)]^2\,dx.$$

Bibliography

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E. LINDELÖF: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).