旋轉曲面

如果曲线由参数方程${\displaystyle x(t)}$、${\displaystyle y(t)}$给出，其中${\displaystyle a<t<b}$，且旋转轴是${\displaystyle y}$轴，则旋转曲面${\displaystyle A}$的面积由以下的积分给出：

${\displaystyle A=2\pi \int _{a}^{b}x(t)\ {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,}$

条件是${\displaystyle x(t)}$非负。这个公式与古尔丁定理是等价的。

${\displaystyle \left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}$

来自勾股定理，表示曲线的一小段弧，像弧长的公式那样。${\displaystyle 2\pi x(t)}$是这一小段的（重心的）路径。

如果曲线的方程是y = f(x)，a ≤ x ≤ b，则积分变为：

${\displaystyle A=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx}$（绕着x轴旋转），

${\displaystyle A=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dx}{dy}}\right)^{2}}}\,dy}$（绕着y轴旋转）。

这可以由以上的公式推出。

例如，单位半径的球面由曲线x(t) = sin(t)，y(t) = cos(t)旋转而得，其中${\displaystyle 0<t<\pi }$。所以，它的面积为：

${\displaystyle A=2\pi \int _{0}^{\pi }\sin(t){\sqrt {\left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}}\,dt=2\pi \int _{0}^{\pi }\sin(t)\,dt=4\pi .}$

对于半径为r的圆${\displaystyle y(x)={\sqrt {r^{2}-x^{2}}}}$绕着x轴旋转所得的曲面，

${\displaystyle A=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx}$

${\displaystyle =2\pi \int _{-r}^{r}r\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx}$

${\displaystyle =2\pi \int _{-r}^{r}r\,dx}$

${\displaystyle =4\pi r^{2}\,}$

• 旋转体

• Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 931-937, 1985.
• Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 42, 1980.
• Gray, A. "Surfaces of Revolution." Ch. 20 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 457-480, 1997.
• Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 in Geometry and the Imagination. New York: Chelsea, pp. 7-11, 1999.
• Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, pp. 79-80 and Appendix III, 1992.