# area of surface of revolution

A *surface of revolution ^{}* is a 3D surface, generated when an arc is rotated fully around a straight line.

The general surface of revolution is obtained when the arc is rotated about an arbitrary axis. If one chooses Cartesian coordinates^{}, and specializes to the case of a surface of revolution generated by rotating about the $x$-axis a curve described by $y$ in the interval^{} $[a,b]$, its area can be calculated by the formula

$$A=2\pi {\int}_{a}^{b}y\sqrt{1+{\left(\frac{dy}{dx}\right)}^{2}}\mathit{d}x$$ |

Similarly, if the curve is rotated about the $y$-axis rather than the $x$-axis, one has the following formula:

$$A=2\pi {\int}_{a}^{b}x\sqrt{1+{\left(\frac{dx}{dy}\right)}^{2}}\mathit{d}y$$ |

The general formula is most often seen with parametric coordinates. If $x(t)$ and $y(t)$ describe the curve, and $x(t)$ is always positive or zero, then the area of the general surface of revolution $A$ in the interval $[a,b]$ can be calulated by the formula

$$A=2\pi {\int}_{a}^{b}y\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}\mathit{d}t$$ |

To obtain a specific surface of revolution, translation^{} or rotation can be used to move an arc before revolving it around an axis. For example, the specific surface of revolution around the line $y=s$ can be found by replacing $y$ with $y-s$, moving the arc towards the $x$-axis so $y=s$ lies on it. Now, the surface of revolution can be found using one of the formulae above.

In this specific case, replacing $y$ with $y=s$, the area of a surface of revolution is found using the formula

$$A=2\pi {\int}_{a}^{b}(y-s)\sqrt{{\left(\frac{dy}{dx}\right)}^{2}}\mathit{d}y$$ |

Title | area of surface of revolution |
---|---|

Canonical name | AreaOfSurfaceOfRevolution |

Date of creation | 2014-07-24 18:36:37 |

Last modified on | 2014-07-24 18:36:37 |

Owner | rspuzio (6075) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | rspuzio (2872) |

Entry type | Topic |

Classification | msc 53A05 |

Classification | msc 26B15 |

Synonym | area of revolution |

Synonym | surface area of revolution |

Related topic | SurfaceOfRevolution2 |

Related topic | VolumeOfSolidOfRevolution |