# 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 $x=a$, $x=b\phantom{\rule{veryverythickmathspace}{0ex}}(>a)$, and a continuous^{} curve $y=f(x)$.

Between the bounds $a$ anb $b$ we fit a sequence of points ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n-1}$ 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 $$ of the union of the cylinders generated by the inscribed rectangles.

Now it is apparent that

$${V}_{>}=\pi [{M}_{1}^{2}({x}_{1}-a)+{M}_{2}^{2}({x}_{2}-{x}_{1})+\mathrm{\dots}+{M}_{n}^{2}(b-{x}_{n-1})],$$ |

$$ |

where ${M}_{1},{M}_{2},\mathrm{\dots},{M}_{n}$ are the greatest and ${m}_{1},{m}_{2},\mathrm{\dots},{m}_{n}$ 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

$$ |

and this is true for any $$ 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 ${\int}_{a}^{b}}\pi {[f(x)]}^{2}\mathit{d}x.$ Therefore the volume of the given solid of revolution can be obtained from

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

## References

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

Title | volume of solid of revolution |
---|---|

Canonical name | VolumeOfSolidOfRevolution |

Date of creation | 2013-03-22 17:20:12 |

Last modified on | 2013-03-22 17:20:12 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 51M25 |

Related topic | PappussTheoremForSurfacesOfRevolution |

Related topic | SurfaceOfRevolution |

Related topic | VolumeAsIntegral |