Applications of Integrals

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

几何应用

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

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

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

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

Examples

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

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

Examples

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

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

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

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

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

符号	类型	读音/说明	在本文中的含义
$\int$	数学符号	integral	定积分符号
$	f(x)	$	符号
$\pi$	希腊字母	Pi（派）	圆周率