Line Integrals with Respect to Specific Variables

# 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}