Line Integrals on Piecewise Smooth Curves
Recall from the Line Integrals page that if $z = f(x, y)$ is a two variable real-valued function and if the smooth curve $C$ is given parametrically by $x = x(t)$ and $y = y(t)$ for $a ≤ t ≤ b$, then the line integral of $f$ along $C$ is given by:(1)
Now suppose instead that the curve $C$ is not smooth such as the one illustrated below:
Geometrically, the curve $C$ is not smooth because $C$ has a sharp point.
Now suppose that $C$ is actually a piecewise smooth curve, that is, $C$ is the union of a finite number $n$ of smooth curves $C_1$, $C_2$, …, $C_n$ as illustrated below:
Then we can still compute the line integral of $f$ along $C$ as the sum of the line integrals of $f$ along $C_1$, $C_2$, …, $C_n$, that is:(2)