# Line Integral

1. Jul 29, 2007

### theperthvan

What does line integral really mean, what is it doing?

Say you have a function f(x,y,z) and you integrate it w.r.t. arc length along some curve C.
Is this like finding the area under C over f? Like if you are walking along C, and the vertical area covered below you is the integral?

It's hard to say what I mean, but is this correct?

Cheers,

2. Jul 29, 2007

### Kummer

Here is an application. Let $$f(x,y,z)$$ represent the mass at every given point. Then $$\int_C f(x,y,z) ds$$ along a rectifiable curve $$C$$ is the total mass of the string/wire (which is represented by the curve).

3. Jul 29, 2007

### sam1

Say you have z = f(x,y). Then $$\int_C f(x,y) ds$$ represents the area of the sheet that is traces out. That is, the line represented by C in the xy plane, connect it to the surface f(x,y), and the line integral represents the area of this sheet

4. Jul 30, 2007

### theperthvan

oh right. that makes more sense. cheers,

5. Jul 30, 2007

### nicktacik

If you are familiar with basic physics, then

$$\int_C{\vec{F}\cdot\ d\vec{r}$$

This line integral represents the work done by the force F along the path C.

$$\int_a^b{\vec{E}\cdot\ d\vec{r}$$

This line integral represents the potential difference (voltage) between points b and a, where E is the electric field.

6. Jul 30, 2007

### jbowers9

I thought the line integral in its mathematical sense represented the "length" of the line between the points of integration.

7. Jul 31, 2007

### Gib Z

Nope you are thinking of arclength. The arclength of an integrable function f(x) over [a,b] is given by $$\int^b_a \sqrt{ 1+ (f'(x))^2} dx$$

8. Aug 2, 2007

### jbowers9

OK, thank you.

9. Aug 3, 2007

### Kummer

It can represent arc length. If $$\bold{R} = x(t)\bold{i}+y(t)\bold{j}$$ (for smooth functions) then $$\int_C 1 \ ds = \int_a^b \sqrt{[x'(t)]^2+[y'(t)]^2} dt$$ where $$C$$ is the path obtained from $$\bold{R}$$.