二维空间

即平面几何

平面几何公式

平面图形面积$S=\begin{cases}\int_{a}^{b}\left|y_1\left(x\right)-y_2\left(x\right)\right|\mathrm{d}x\\ \dfrac12\int_{\alpha}^{\beta}\left|r_1^2\left(\theta\right)-r_2^2\left(\theta\right)\right|d\theta\end{cases}$

旋转体的体积$V=\begin{cases}\pi\int_{a}^{b}\left|y_1^2\left(x\right)-y_2^2\left(x\right)\right|\mathrm{d}x\\ 2\pi\int_{a}^{b}x\left|y_1\left(x\right)-y_2\left(x\right)\right|\mathrm{d}x\end{cases}$

弧微分$\mathrm{ds}=\sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}\mathrm{dt}=\sqrt{r_{\theta}^2+r_{\theta}^{\prime2}}d\theta=\sqrt{1+f_{x}^{\prime2}}dx=\sqrt{x_{t}^{\prime2}+y_{t}^{\prime2}}dt$

平面曲线弧长$s=\int ds=\int_{a}^{b}\sqrt{1+f_{x}^{\prime2}}dx=\int_{\alpha}^{\beta}\sqrt{x_{t}^{\prime2}+y_{t}^{\prime2}}dt=\int_{\alpha}^{\beta}\sqrt{r_{\theta}^2+r_{\theta}^{\prime2}}d\theta$

平面图形形心$\bar{x}=\dfrac{\iint_{D}xd\sigma}{\iint_{D}d\sigma}$

曲率$K=\dfrac{d\theta}{d\rho} = \dfrac{|y''|}{\rho^3}$

曲率半径$R=\dfrac{1}{K}$

旋转曲面表面积$S=2\pi\int_{a}^{b}\left|y\left(x\right)\right|\sqrt{1+\left\lbrack y^{\prime}\left(x\right)\right\rbrack^2}\mathrm{d}x=2\pi\int_{\alpha}^{\beta}\left|y\left(t\right)\right|\sqrt{\left\lbrack x^{\prime}\left(t\right)\right\rbrack^2+\left\lbrack y^{\prime}\left(t\right)\right\rbrack^2}\mathrm{d}t$

扇形面积

三角形面积

椭圆

双曲线

抛物线

圆

物体体积$\iiint_{\Omega}dv=V$

区域面积$A=\iint_{D}d\sigma$