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# 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 $[a,\,x_{1}]$, $[x_{1},\,x_{2}]$, …, $[x_{{n-1}},\,b]$. The volume $V$ of the solid of revolution thus satisfies

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

and this is true for any 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

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

# References

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

## Mathematics Subject Classification

51M25*no label found*

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