If a smooth curve $x=f(t),y=g(t),a\le t\le b$, is traversed exactly once as $t$ increases from $a$ to $b$, the curve's length is

$$L={\int }_a^{b}ds={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$$

The arc length of curve in ${\mathbb{R}}^3$ can be defined the same way, assuming $z=h(t)$:

$$L={\int }_a^{b}ds={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2+{\left(\frac{dz}{dt}\right)}^2}dt$$

If $f$ is a continuous function of two variables $x=x(t),y=y(t)$ whose domain includes the smooth curve $C$, then the line integral can be evaluated as:

$${\int }_{C}f(x,y)ds={\int }_a^{b}f(x(t),y(t))\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$$

When $P(x,y)$ and $Q(x,y)$ are continuous and $C$ is smooth, we can calculate the value of the line integral by expressing $P,Q,dx$ and $dy$ in terms of any parametric representation of $C$ and evaluating the resulting definite integral:

$$\begin{array}{cc}{\int }_{C}P(x,y)dx+Q(x,y)dy={\int }_{\alpha }^{\beta }\{P[x(t),y(t)]\frac{dx}{dt}+Q[x(t),y(t)]\frac{dy}{dt}\}dt.& \text{(4.7)}\end{array}$$

All parameterizations of the curve lead to the same value.

When a line integral is known to be independent of path, and its value is required along some curve with initial point $A$ and final point $B$, we can either replace the given curve with a simpler curve, or, take the difference in the values of the function $\varphi (x,y)$ at $B$ and $A$.

$${\int }_{C}P(x,y)dx+Q(x,y)dy={\{\varphi (x,y)\}}_A^B=\varphi (x_B,y_B)-\varphi (x_A,y_A).$$

If a line integral is independent of path in a domain $D$, and $C$ is a closed, piecewise smooth curve in $D$ that contains only points of $D$ in its interior, its value is zero:

$${\oint }_{C}Pdx+Qdy=0$$