Units

# Unit 12: Line Integrals

## The Concept

For a single-variable integral in 2D, $\int_a^b f(x)\,dx$, we integrate function $f(x)$ along $x$ in 2D and it represents the area inbetween the curve, $y=f(x)$, and a segment of $x$-axis from $a$ to $b$.

A line integral in 3D shares a similar idea to a single-variable integral in 2D. A line integral, $\int_C f(x,y)\,ds$, integrates the surface function, $z=f(x,y)$, along a 2D curve segment $C$ on the $xy$-plane, instead of $x$ on the $x$-axis or $y$ on the $y$-axis alone. This line segment, $C$, is described by a vector function, $r(t)=\langle x(t), y(t) \rangle$, where $t=a$ and $t=b$ map the start point and end point of $C$, respectively. The differential element, $ds$, represents the change of arc length of curve $C$, i.e., $ds=\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,\,dt$. Thus the line integral can be evaluated by the following single integral:

$\int_C f(x,y)\,ds = \int_a^b f(x(t),y(t))\, \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,\,dt$.

The value of $\int_C f(x,y)\,ds$ is the area of the “wall”, “fence” or “curtain” whose base is the 2D curve $C$ on the $xy$-plane and and whose height is given by the function $z=f(x,y)$.

The concept of line integral can be extended to high dimensions. For example, $\int_C f(x,y,z)\,ds$ integrates the function with three variables, $w=f(x,y,z)$, along a 3D curve C that is parameterized by $r(t) = \langle x(t),y(t),z(t) \rangle$. It can be evaluated by a single integral as well, that is,

$\int_C f(x,y,z)\,ds = \int_a^b f(x(t),y(t),z(t))\, \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2+(\frac{dz}{dt})^2} \,\,dt$.

## The Plot

Now, you should engage with the 3D plot below to understand tangent planes . Follow the steps below to apply changes to the plot and observe the effects:

1. Input the function $f(x, y)$.
2. Adjust the 2D curve $C$ on the $xy$-plane.
3. Adjust the number of rectangular subareas, $n$.
4. The estimation of the line integral is shown. The larger $n$ is, the better the estimation is.

## Self-Checking Questions

Check your understanding by solving the following questions:

1. Find the value of integral $\int_C(x^2+y^2)\, ds$, where $C$ is part of the helix parameterized by $r(t)=\langle cos t, sin t \rangle, 0 \leq t \leq 2$.
2. Evaluate $\int_C \frac{1}{x^2+y^2} \, ds$ , over the line segment from $(1,1)$ to $(3,0)$.

Use the graph to find the answers to these questions. 