Unit 12: Line Integrals

The Concept

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

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

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

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

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

[latex]\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[/latex].

The Plot

Now, you should engage with the 3D plot below to understand tangent planes ^[1]. Follow the steps below to apply changes to the plot and observe the effects:

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