Line Integrals Review

We will now review some of the recent content regarding line integrals.

• Recall from the Line Integrals page that if $z = f(x, y)$ is a two variable real-valued function and if $C$ is a smooth curve given parametrically by $x = x(t)$ and $y = y(t)$ for $a ≤ t ≤ b$, then the Line Integral of $f$ Along $C$ is:

(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}

• Geometrically, a line integral of a two variable real-valued function $f$ along a curve $C$ is the area of either side of the "dropped curtain" hung on the surface generated by $f$ and draped onto the curve $C$ in the $xy$-plane.

• Furthermore, if $w = f(x, y, z)$ is a three variable real-valued function and if $C$ is a smooth curve given parametrically by $x = x(t)$, $y = y(t)$ and $z = z(t)$ for $a ≤ t ≤ b$, then the line integral of $f$ along $C$ is:

(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 \end{align}

• More compactly, we can write both of these line integral formulas as:

(3)

\begin{align} \quad \int_a^b f(\vec{r}(t)) \| r'(t) \| \: dt \end{align}

• We also saw that the parameterization of the curve $C$ did not affect the value of these line integrals, that is, if $\vec{r}(t)$ is a parameterization of the curve $C$ for $a ≤ t ≤ b$ and $\vec{r^*}(u)$ is also a parameterization of $C$ for $\alpha ≤ u ≤ \beta$ then:

(4)

\begin{align} \quad \int_a^b f(\vec{r}(t)) \| \vec{r'}(t) \| \: dt = \int_{\alpha}^{\beta} f(\vec{r^{*}}(u)) \| \vec{r^*{'}}(u) \| \: du \end{align}

• On the Line Integrals on Piecewise Smooth Curves page we saw that if $C$ was not a smooth curve and instead $C$ was a finite union of piecewise smooth curves $C_1$, $C_2$, …, $C_n$ then we can still compute the line integral of a function $f$ along $C$ and:

(5)

\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 \end{align}

• On the Line Integrals with Respect to Specific Variables page we saw that we could also take line integrals with respect to specific variables. For example, if $z = f(x, y)$ is a two variable real-valued function and $C$ is a 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:

(6)

\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 $y$ is:

(7)

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

• We also saw from the Line Integrals of Vector Fields page that if $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a continuous vector field on $\mathbb{R}^3$ and if $x = x(t)$, $y = y(t)$, and $z = z(t)$ parameterize a smooth curve $C$ for $a ≤ t ≤ b$, then the **Line Integral of $\mathbf{F}$ Along $C$ is:

(8)

\begin{align} \quad \int_C \mathbf{F} \cdot d \vec{r} = \int_C P(x, y, z) \: dx + Q(x, y, z) \: dy + R(x, y, z) \: dz \\ \quad \int_C \mathbf{F} \cdot d \vec{r} = \int_a^b \left [ P(x(t), y(t), z(t))x'(t) + Q(x(t), y(t), z(t))y'(t) + R(x(t), y(t), z(t))z'(t) \right ]\: dt \end{align}

• Another notation for a line integral of $\mathbf{F}$ along the curve $C$ is $\int_C \mathbf{F} \cdot \hat{T} \: ds$ which is also known as the Line Integral of the Tangent Component of $\mathbf{F}$.

• If the direction of the orientation of $C$ changes, then the sign of the line integral of $\mathbf{F}$ along $C$ changes signs, that is:

(9)

\begin{align} \quad \int_{-C} \mathbf{F} \cdot d \vec{r} = - \int_C \mathbf{F} \cdot d \vec{r} \end{align}

• Lastly, if $C$ is a closed curve then we sometimes right a little circle on the integral symbol to denote this. Thus, $\oint_C \mathbf{F} \cdot d \vec{r}$ should be regarded as the line integral of $\mathbf{F}$ along the closed curve $C$.

• On the The Fundamental Theorem for Line Integrals we saw The Fundamental Theorem for Line Integrals, also known as The Gradient Theorem which says that if $C$ is a smooth curve parameterized by $\vec{r}(t)$ for $a ≤ t ≤ b$ and if $f$ is a differentiable function whose gradient $\nabla f$ is continuous then:

(10)

\begin{align} \quad \int_C \nabla f \cdot d \vec{r} = f(\vec{r}(b)) - f(\vec{r}(a)) \end{align}

• Therefore if $\mathbf{F}$ is a conservative vector field whose potential is $f$, that is $\mathbf{F} = \nabla f$ then evaluating line integrals is as simple as finding the values of $f$ at the end points of the curve $C$.