Line Integrals Examples 3

Recall from the Line Integrals page that if $z = f(x, y)$ is a two variable real-valued function and $C$ is a smooth plane curve defined by the parametric equations $x = x(t)$ and $y = y(t)$ 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 = \int_a^b f(\vec{r}(t)) \| \vec{r''}(t) \| \: dt \end{align}

Similarly if $z = f(x, y, z)$ is a three variable real-valued function and $C$ is a smooth space curve defined by the parametric equations $x = x(t)$, $y = y(t)$ and $z = z(t)$ then the line integral of $f$ along $C$ is given by:

(2)

\begin{align} \quad \int_C f(x, y, z) \: ds = \int_a^b f(x(t), y(t), z(t)) \sqrt{\left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right)^2 + \left ( \frac{dz}{dt} \right )^2} \: dt = \int_a^b f(\vec{r}(t)) \| \vec{r'}(t) \| \: dt \end{align}

We will now look at some more examples of computing line integrals.

Example 1

Evaluate $\int_C x^2 \: ds$ along the line of intersection of $x - y + z = 0$ and $x + y + 2z = 0$ from the point $(0, 0, 0)$ to $(3, 1, -2)$.

Note that $x - y + z = 0$ and $x + y + 2z = 0$ are both planes and either intersect at a straight line, do not intersect, or coincide. It's not hard to see that these planes in fact intersect at a straight line. We can thus parameterize the line segment from $(0, 0, 0)$ to $(3, 1, -2)$ for $0 ≤ t ≤ 1$ as:

(3)

\begin{align} \quad \vec{r}(t) = (1 - t)(0, 0, 0) + t(3, 1, -2) = (3t, t, -2t) \end{align}

Thus:

(4)

\begin{align} \quad \int_C x^2 \: ds = \int_0^1 (3t)^2 \sqrt{(3)^2 + (1)^2 + (-2)^2} \: dt \\ \quad \int_C x^2 \: ds = \int_0^1 9 \sqrt{14} t^2 \: dt \\ \quad \int_C x^2 \: ds = 9 \sqrt{14} \left [ \frac{t^3}{3} \right ]_{t=0}^{t=1} \\ \quad \int_C x^2 \: ds = 3 \sqrt{14} \end{align}