LINE INTEGRALS | Improve Society

Fig. 6-2

Let C be a curve in the xy plane which connects points $A (a_1, b_1)$ and $B (a_2, b_2)$ , (see Fig. 6-2). Let $P(x, y)$ and $Q(x, y)$ be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by $(x_1, y_1),\ (x_2, y_2),\ \dots,\ (x_{n-1}, y_{n-1})$ . Call $\Delta x_k = x_k - x_{k-1}$ and $\Delta y_k = y_k - y_{k-1},\ k = 1,\ 2,\ \dots\ n$ and suppose that points $(\xi_k, \eta_k)$ are chosen so that they are situated on C between points $(x_{k-1}, y_{k-1})$ and $(x_k, y_k)$ . Form the sum
$\displaystyle \sum_{k=1}^{n}\{P(\xi_k, \eta_k)\Delta x_k + Q(\xi_k, \eta_k)\Delta y_k\}\cdots(13)$
The limit of this sum as $n\rightarrow\infty$ in such a way that all quantities $\Delta x_k,\ \Delta y$ approaches zero, if such limit exists, is called a line integral along C and is denoted by

$\displaystyle \int_C \left[ P(x, y)dx + Q(x, y)dy \right]$ or $\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)}\left[ Pdx + Qdy \right]\cdots(14)$

The limit does exist if P and Q are continuous (or piecewise continuous) at all points of C. The value of the integral depends in general on P, Q, the particular curve C, and on the limits $(a_1, b_1)$ and $(a_2, b_2)$ .

In an exactly analogous manner one may define a line integral along a curve C in three dimensional space as
$\displaystyle \lim\limits_{n \rightarrow\infty}\sum_{k=1}^{n}\left\{ A_1(\xi_k, \eta_k, \zeta_k)\Delta x_k + A_2(\xi_k, \eta_k, \zeta_k)\Delta y_k + A_3(\xi_k, \eta_k, \zeta_k)\Delta z_k \right\} \\ = \int_C \left[ A_1dx + A_2dy + A_3dz \right] \cdots(15)$
where $A_1$ , $A_2$ and $A_3$ are functions of $x$ , $y$ and $z$ .

Other types of line integrals, depending on particular curves, can be defined. For example, if $\Delta s_k$ denotes the arc length along curve C in the above figure between points $(x_k, y_k)$ and $(x_{k+1}, y_{k+1})$ , then
$\displaystyle \lim\limits_{n \rightarrow \infty} \sum_{k=1}^{n} U(\xi_k, \eta_k)\Delta s_k = \int_C U(x, y)ds\cdots(16)$
is called the line integral of $U(x, y)$ along curve C. Extensions to three (or higher) dimensions are possible.