Line Integrals on Piecewise Smooth Curves

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)
\begin{align} \quad \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 \end{align}

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)
\begin{align} \quad \int_C f(x, y) \: ds = \int_{C_1} f(x, y) \: ds + \int_{C_2} f(x, y) \: ds + ... + \int_{C_n} f(x, y) \: ds \\ \quad \int_C f(x, y) \: ds = \sum_{i=1}^{n} \int_{C_i} f(x, y) \: ds \end{align}