Line Integrals with Respect to Specific Variables

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 $y = y(t)$ for $a ≤ t ≤ b$ then the line integral of $f$ along $C$ is given by the formula:

(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 $w = f(x, y, z)$ is a three variable real-valued function and $C$ is a smooth space curve defined by the parametric equation $x = x(t)$, $y = y(t)$, and $z = z(t)$ for $a ≤ t ≤ b$ then the line integral of $f$ along $C$ is given by the formula:

(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}

Another type of line integral that we can evaluate are known as line integrals with respect to one of the independent variables.

Line Integrals with Respect to Specific Variables for Functions of Two Variables

Definition: If $z = f(x, y)$ is a two variable real-valued function and $C$ is a smooth plane curve parameterized as $x = x(t)$ and $y = y(t)$ for $a ≤ t ≤ b$ then the Line Integral of $f$ Along $C$ with Respect to $x$ is $\int_C f(x, y) \: dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*, y_i^*) \Delta x_i$, and the Line Integral of $f$ Along $C$ with Respect to $y$ is $\int_C f(x, y) \: dy = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*, y_i^*) \Delta y_i$.

The line integral of $f$ along $C$ with respect to $x$ can be evaluated using the following formula:

(3)

\begin{align} \quad \int_C f(x, y) \: dx = \int_a^b f(x(t), y(t)) x'(t) \: dt \end{align}

Similarly the line integral of $f$ along $C$ with respect to $y$ can be evaluated using the following formula:

(4)

\begin{align} \quad \int_C f(x, y) \: dy = \int_a^b f(x(t), y(t)) y'(t) \ dt \end{align}

Sometimes the line integrals of $f$ along $C$ with respect to the variables $x$ and $y$ occur together and in such case, we will often times use the following abbreviation to denote their sum:

(5)

\begin{align} \quad \int_C P(x, y) \: dx + Q(x, y) \: dy = \int_C P(x, y) \: dx + \int_C Q(x, y) \: dy \end{align}

Line Integrals with Respect to Specific Variables for Functions of Three Variables

Definition: If $w = f(x, y, z)$ is a three variable real-valued function and $C$ is a smooth space curve parameterized as $x = x(t)$, $y = y(t)$, and $z = z(t)$ for $a ≤ t ≤ b$ then the Line Integral of $f$ Along $C$ with Respect to $x$ is $\int_C f(x, y, z) \: dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*, y_i^*, z_i^*) \Delta x_i$, the Line Integral of $f$ Along $C$ with Respect to $y$ is $\int_C f(x, y, z) \: dy = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*, y_i^*, z_i^*) \Delta y_i$, and the Line Integral of $f$ Along $C$ with Respect to $z$ is $\int_C f(x, y, z) \: dz = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*, y_i^*, z_i^*) \Delta z_i$.

The line integral of $f$ along $C$ with respect to $x$ can be evaluated with the following formula:

(6)

\begin{align} \quad \int_C f(x, y, z) \: dx = \int_a^b f(x(t), y(t), z(t)) x'(t) \: dt \end{align}

The line integral of $f$ along $C$ with respect to $y$ can be evaluated with the following formula:

(7)

\begin{align} \quad \int_C f(x, y, z) \: dy = \int_a^b f(x(t), y(t), z(t)) y'(t) \: dt \end{align}

The line integral of $f$ along $C$ with respect to $z$ can be evaluated with the following formula:

(8)

\begin{align} \quad \int_C f(x, y, z) \: dz = \int_a^b f(x(t), y(t), z(t)) z'(t) \: dt \end{align}

Similarly, the line integrals of $f$ along $C$ with respect to $x$, $y$, and $z$ can occur together in such case we use the following notation for abbreviation:

(9)

\begin{align} \quad \int_C P(x, y, z) \: dx + Q(x, y, z) \: dy + R(x, y, z) \: dz = \int_C P(x, y, z) \: dx + \int_C Q(x, y, z) \: dy + \int_C R(x, y, z) \: dz \end{align}