Applications of Integrals

Integrals are powerful in both theory and practice. Here are geometric and physical applications.

几何应用

平面图形面积

If $f(x)$ is continuous on $[a,b]$, the area with the $x$-axis is

$S = \int_a^b |f(x)| \, dx$

Absolute value handles negative regions.

Examples

1. $y=x^2$ on $[0,1]$: $S=\int_0^1 x^2 dx = \tfrac13$.
2. $y=\sin x$ on $[0,\pi]$: $S=\int_0^{\pi}\sin x dx = 2$.
3. $y=x^3 - x$ on $[-1,1]$: split sign; result $\tfrac12$.

平面曲线弧长

If $y=f(x)$ is $C^1$ on $[a,b]$,

$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$

Example $y=x^2$ on $[0,1]$: $L=\tfrac12\bigl(\sqrt{5}+\tfrac12\ln(2+\sqrt5)\bigr)$.

旋转体体积

For $y=f(x)$ continuous on $[a,b]$, revolving about $x$-axis:

$V = \pi \int_a^b [f(x)]^2 \, dx$

Examples

1. $y=x^2$, $[0,1]$: $V=\dfrac{\pi}{5}$.
2. $y=\sin x$, $[0,\pi]$: $V=\dfrac{\pi^2}{2}$.

物理应用

功（Work）

Force $F(x)$ along $x$ from $a$ to $b$:

$W = \int_a^b F(x) \, dx$

Examples

1. Spring $F(x)=kx$: $W=\dfrac{ka^2}{2}$ from $0$ to $a$.
2. Gravity $F(r)=\dfrac{GMm}{r^2}$: $W = GMm\left(\dfrac{1}{a}-\dfrac{1}{b}\right)$.

质心（Center of mass）

For thin plate with density $\rho(x)$ on $[a,b]$,

$\bar{x} = \frac{\int_a^b x\,\rho(x)\,dx}{\int_a^b \rho(x)\,dx}$

电荷分布、电场

Continuous charge density $\lambda(x)$ on a wire: total charge $Q=\int \lambda(x)\,dx$; fields computed via integrals.

概率

Continuous random variable with density $f(x)$: $P(X\in [a,b])=\int_a^b f(x)\,dx$.

Summary

本文出现的符号

中英对照