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# 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

$A=2\pi\int_{{a}}^{{b}}y\,\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx$ |

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

$A=2\pi\int_{{a}}^{{b}}x\,\sqrt{1+\left(\frac{dx}{dy}\right)^{2}}\,dy$ |

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

$A=2\pi\int_{{a}}^{{b}}y\,\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{% dt}\right)^{2}}\,dt$ |

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

$A=2\pi\int_{{a}}^{{b}}(y-s)\sqrt{\left(\frac{dy}{dx}\right)^{2}}\,dy$ |

## Mathematics Subject Classification

53A05*no label found*26B15

*no label found*

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