积分的应用

计算体积

切片法

旋转轴为\(y=h\)

\[V=\int_a^b\pi (y-h)^2 dx\]

1.11

壳法

旋转轴为\(x=h\)

\[V=\int_a^b2\pi(x-h)ydx\]

1.12

椎体体积

椎体体积

\[V=\int_0^hA(x)dx=\frac{1}{3}Ah\]

1.10

Proof:
\(\begin{align} \frac{x}{l}=\frac{h}{L}\\ \Rightarrow \frac{A}{A(x)}=&(\frac{L}{l})^2\\ \Rightarrow A(x)=\frac{Ax^2}{h^2}\\ V=&\int _0^hA(x)dx\\ =&\int _0^h\frac{A}{h^2}x^2dx\\ =&\frac{1}{3}Ah \end{align}\)

计算弧长

\[L=\int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx\\ L=\int_{t_0}^{t_1} \sqrt{(\frac{dy}{dt})^2+(\frac{dx}{dt})^2}dx\\ L=\int_{\theta_0}^{\theta_1} \sqrt{f(\theta)^2+f'(\theta)^2}dx\]

1.14

计算旋转体表面积

\[S=\int_a^b2\pi y\sqrt{1+(\frac{dy}{dx})^2}dx\\ S=\int_{t_0}^{t_1}2\pi y\sqrt{(\frac{dy}{dt})^2+(\frac{dx}{dt})^2}dt\]

1.13