Concise Notes on Integrals - Types of Curve Integrals of the First Kind

Types of curve integrals of the first kind

一、Classification of curve integrals of the first kind

01 Plane curve integrals of the first kind

若 ${\Gamma }_{AB}\subset {\mathrm{R}}^2$,$P(x,y)\in {\Gamma }_{AB}$,

${\int }_{{\Gamma }_{AB}}f(P)ds={\int }_{{\Gamma }_{AB}}f(x,y)ds$ It is called the curve integral of the first kind in the plane.

02 Curve integrals of the first kind in space

若 ${\Gamma }_{AB}\subset {\mathrm{R}}^3$,$P(x,y,z)\in {\Gamma }_{AB}$,

${\int }_{{\Gamma }_{AB}}f(P)ds={\int }_{{\Gamma }_{AB}}f(x,y,z)ds$ It is called the curve integral of the first kind in space.

Curve integrals of the first kind can be summed in planes空间The arc length of the curve.

二、Calculation of curve integrals of the first kind

01 Plane curve integrals of the first kind

(1) 一元函数型

形式：${\Gamma }_{AB}:y=\phi (x),x\in [a,b]$ (x=x) 特殊的参数方程

$\phi ’(x)$ 连续（以后默认）
$${\int }_{{\Gamma }_{AB}}f(x,y)ds={\int }_a^{b}f(x,\phi (x))\sqrt{1+{\phi }^{\prime 2}(x)}dx$$

(2) 一元反函数型

形式：${\Gamma }_{AB}:x=\psi (y),y\in [c,d],\psi ’(x)$ 连续 ( $y=y$ ) 特殊的参数方程
$${\int }_{{\Gamma }_{AB}}f(x,y)ds={\int }_c^{d}f(\psi (y),y)\sqrt{1+{\psi }^{\prime 2}(y)}dy$$

(3) 极坐标型

形式：${\Gamma }_{AB}:r=r(\theta ),\theta \in [\alpha ,\beta ],r^{\prime }(\theta )$ 连续

$\Rightarrow \{\begin{array}{l}x=r(\theta )\cos \theta \\ y=r(\theta )\sin \theta \end{array}\theta \in [\alpha ,\beta ]$ ,$x^{\prime 2}(\theta )+y^{\prime 2}(\theta )=r^2(\theta )+r^{\prime 2}(\theta )$,Force the construction of parametric equations
$${\int }_{{\Gamma }_{AB}}f(x,y)ds={\int }_{\alpha }^{\beta }f(r(\theta )\cos \theta ,r(\theta )\sin \theta )\sqrt{r^2(\theta )+r^{\prime 2}(\theta )}d\theta$$

(4) 反极坐标型

形式：${\Gamma }_{AB}:\theta =\theta (r),r\in [a,b],{\theta }^{\prime }(r)$ 连续

$\Rightarrow \{\begin{array}{l}x=r\cos \theta (r)\\ y=r\sin \theta (r)\end{array}r\in [a,b]$

Do not make a unified form of the formula,具体题目具体分析.

好的思路：转化为 $r=r(\theta )$ 或者关于x,y的方程.

02 Curve integrals of the first kind in space

若 ${\Gamma }_{AB}:\{\begin{array}{l}x=x(t)\\ y=y(t)\\ z=z(t)\end{array}t\in [\alpha ,\beta ]$,则有
$${\int }_{{\Gamma }_{AB}}f(x,y,z)ds={\int }_{\alpha }^{\beta }f(x(t),y(t),z(t))\sqrt{x^{\prime 2}(t)+y^{\prime 2}(t)+z^{\prime 2}(t)}dt$$

The multiintegrated integrand cannot generally be simplified,Because it satisfies the inequality,The curve-surface integral can often be simplified,Because it satisfies the equation.