求曲线本身属性

XantC

曲线长度

\[ s=\begin{cases} \displaystyle\int_{x_a}^{x_b}\sqrt{1+(\frac{\mathrm{d}y}{\mathrm{d}x})^2}\,\mathrm{d}x \\ \displaystyle\int_{y_a}^{y_b}\sqrt{1+(\frac{\mathrm{d}x}{\mathrm{d}y})^2}\,\mathrm{d}y \\ \displaystyle\int_{t_a}^{t_b}\sqrt{(\frac{\mathrm{d}x}{\mathrm{d}t})^2+(\frac{\mathrm{d}y}{\mathrm{d}t})^2}\,\mathrm{d}t \\ \end{cases} \]

推导

\[ (\delta s)^2\approx(\delta x)^2+(\delta y)^2 \]

\[ \Rightarrow \frac{\mathrm{d}s}{\mathrm{d}x}=\sqrt{1+(\frac{\mathrm{d}y}{\mathrm{d}x})^2} \]

\[ \Rightarrow s=\int_{x_a}^{x_b}\sqrt{1+(\frac{\mathrm{d}y}{\mathrm{d}x})^2}\,\mathrm{d}x \]

曲线旋转体

体积

\[ \begin{align*} \text{绕x轴旋转：} V=\pi\int_{x_a}^{x_b}y^2\mathrm{d}x \\ \text{绕y轴旋转：} V=\pi\int_{y_a}^{y_b}x^2\mathrm{d}y \end{align*} \]

侧表面积

基本公式

\[ \begin{align*} \text{绕x轴旋转：} S=\int2\pi y\mathrm{d}s=\int2\pi y\frac{\mathrm{d}s}{\mathrm{d}x}\mathrm{d}x \\ \text{绕y轴旋转：} S=\int2\pi x\mathrm{d}s=\int2\pi x\frac{\mathrm{d}s}{\mathrm{d}y}\mathrm{d}y \\ \end{align*} \]

用于计算的公式

\[ 以绕x轴旋转为例： \]

\[ S=\begin{cases} \displaystyle 2\pi\int_{x_a}^{x_b}y\sqrt{1+(\frac{\mathrm{d}y}{\mathrm{d}x})^2}\,\mathrm{d}x \\ \displaystyle 2\pi\int_{y_a}^{y_b}y\sqrt{1+(\frac{\mathrm{d}x}{\mathrm{d}y})^2}\,\mathrm{d}y \\ \displaystyle 2\pi\int_{t_a}^{t_b}y\sqrt{(\frac{\mathrm{d}x}{\mathrm{d}t})^2+(\frac{\mathrm{d}y}{\mathrm{d}t})^2}\,\mathrm{d}t \\ \end{cases} \]

推导

\[ 圆台侧表面积公式为S=\pi(r_1+r_2)l \]

\[ \therefore \delta S=\pi(y+(y+\delta y))\delta s \]

\[ \frac{\delta S}{\delta x}=\pi(2y+\delta y)\frac{\delta s}{\delta x} \]

\[ \frac{\mathrm{d}S}{\mathrm{d}x}=2\pi y\frac{\mathrm{d}s}{\mathrm{d}x} \]

\[ \Rightarrow S=\int2\pi y\mathrm{d}s \]