Line Integral

Overview

In Calculus 3, line integrals allow us to integrate functions along a curve in space. They are particularly useful in physics and engineering for calculating work done by a force along a path.

Definition

A line integral of a scalar field $f(x,y,z)$ along a curve $C$ is defined as:

$${\int }_{C}f(x,y,z)ds$$

where $ds$ represents an infinitesimal arc length along the curve $C$.

Intuition

Think of $ds$ as a small segment along the path $C$, and $f(x,y,z)$ as the value of a function (like temperature or density) at each point along this path. The line integral, then, is the total “accumulated” value of $f$ along $C$.

Parametrization

To evaluate a line integral, we often need to parametrize the curve $C$. If $C$ is parametrized by a vector function $\vec{r}(t)=⟨x(t),y(t),z(t)⟩$ over an interval $a\le t\le b$, then:

$$ds=|{\vec{r}}^{\prime }(t)|dt$$

This transforms our line integral into:

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

Parametrization Example

For a curve $C$ along the path $y=x^2$ in the $xy$-plane, we could let $x=t$ and $y=t^2$ for $t$ in some interval $[a,b]$.

Line Integrals of Vector Fields

In many applications, we compute the line integral of a vector field $\vec{F}(x,y,z)$ along a curve $C$:

$${\int }_C\vec{F}\cdot d\vec{r}$$

where $d\vec{r}$ is the differential vector along the curve. If $C$ is parametrized by $\vec{r}(t)$ as before, then $d\vec{r}={\vec{r}}^{\prime }(t)dt$ and we get:

$${\int }_C\vec{F}\cdot d\vec{r}={\int }_a^b\vec{F}(\vec{r}(t))\cdot {\vec{r}}^{\prime }(t)dt$$

Interpretation in Physics

The line integral ${\int }_C\vec{F}\cdot d\vec{r}$ can represent the work done by a force field $\vec{F}$ along a path $C$, where $\vec{F}$ is a vector field representing force and $d\vec{r}$ is the differential displacement vector.

Summary

• Scalar line integrals integrate a scalar function along a curve.
• Vector line integrals calculate the work done by a vector field along a curve.
• Parametrize the curve $C$ to transform the line integral into a form that can be evaluated.