Theory

Introduction

A scalar line integral is the integral of a function over the length of a curve.

A line integral sums the values of a scalar function on space multiplied over the length of some curve in space. This would be an ordinary integral if the curve were stretched out onto an $x$-axis and the function values plotted on the $y$-axis.

Line integral with parametrization $\mathbf{r}(t)$
Suppose $f:{\mathbb{R}}^3\to \mathbb{R}$ is a function on space with values given by $f(x,y,z)$, where $(x,y,z)$ are the coordinates of the input point in space. Suppose a curve $C$ is the image of a parametric curve given with component functions by $\mathbf{r}(t)=(x(t),y(t),z(t))$, and assume $C$ is traversed exactly once by $\mathbf{r}(t)$. Then we compute the line integral of $f$ over $C$ using the formula:

$${\int }_{C}f(x,y,z)ds={\int }_a^{b}f(\mathbf{r}(t))\cdot |{\mathbf{r}}^{\prime }(t)|dt.$$

This formula uses our expression for infinitesimal distance traveled:

$$ds=|{\mathbf{r}}^{\prime }(t)|dt=\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2+z^{\prime }(t{)}^2}dt.$$

Rigorous definition idea
A rigorous definition of the line integral ${\int }_{C}fds$ would divide the curve $C$ into $n$ segments joined at the points $(x_i,y_i,z_i)$, with segment length $\Delta s_i$, then take the Riemann sum:

$$\sum _{i=1}^{n}f(x_i,y_i,z_i)\Delta s_i,$$

and finally take the limit as the number of segments goes to infinity.

Line integral with arc length parametrization
If we start with a curve $\mathbf{r}(s)$ parametrized by arc length, the line integral is very simply given by:

$${\int }_{C}fds={\int }_a^{b}f(\mathbf{r}(s))ds,$$

where $f$ is a scalar function of points in ${\mathbb{R}}^3$.

In fact, the line integral ${\int }_{C}fds$ is sometimes called the line integral with respect to arc length.

Line integral with respect to $x$, $y$, or $z$
A related construction is the line integral with respect to some coordinate. For example, the line integral of $f$ over a curve $C$ with respect to $x$ is given by the formula:

$${\int }_{C}fdx={\int }_a^{b}f(\mathbf{r}(t))\cdot x^{\prime }(t)dt.$$

Sometimes a sum of such integrals is written with shorthand notation as a single integral:

$${\int }_{C}fdx+gdy={\int }_a^{b}f(\mathbf{r}(t))\cdot x^{\prime }(t)+g(\mathbf{r}(t))\cdot y^{\prime }(t)dt.$$

When $C$ is traversed by $\mathbf{r}(t)$, this shorthand can be remembered with formulas using Leibniz notation:

$$dx=x^{\prime }(t)dt,dy=y^{\prime }(t)dt,dz=z^{\prime }(t)dt.$$