What is the meaning of a line integral and how is it calculated?

[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,

 

[Kummer]

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,

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).

 

[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

 

[theperthvan]

oh right. that makes more sense. cheers,

 

[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.

 

[jbowers9]

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

 

[Gib Z]

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

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

 

[jbowers9]

OK, thank you.

 

[Kummer]

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

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}.