查看“︁微积分学/曲面积分”︁的源代码

==例題==
求 <math>x=e^t-t,\;y=4e^{\frac t2},\;0<t<1</math> 繞 y 軸的表面積

<math>
\begin{align}
&
A=\int_C2\pi rdS=\int2\pi xd\sqrt{x^2+y^2}
\\&
=\int2\pi x\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt
\\&
=\int_0^12\pi x\sqrt{\left(e^t-1\right)^2+\left(2e^\frac t2\right)^2}dt
\\&
=\int_0^12\pi x\sqrt{e^{2t}-2e^t+1+4e^t}dt
\\&
=\int_0^12\pi x\sqrt{e^{2t}+2e^t+1}dt
\\&
=\int_0^12\pi\left(e^t-t\right)\left(e^t+1\right)dt
\\&
=2\pi\int_0^1e^{2t}+e^t-te^t-t\,dt
\\&
=2\pi\left[\frac{e^{2t}}2+e^t-\left(te^t-e^t\right)-\frac{t^2}2\right]_0^1
\\&
=2\pi\left[\frac{e^2}2+e-\frac12-\left(\frac12+1+1\right)\right]=\pi\left(e^2+2e-6\right)
\end{align}
</math>