# Surface Integral - or Line Integral?

1. Feb 27, 2010

### bon

1. The problem statement, all variables and given/known data

Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0).

2. Relevant equations

3. The attempt at a solution

So I don't know if this is meant to be a line integral or surface one?

My feeling is that it should be a surface integral over that pentagonal surface..I.e. double integral of F.n dS..where F is (-1,-1,1)..

Firstly, is this right? Secondly, how do I use the fact that the speed of flow is 0.4? Thirdly, how do I find the normal to the plane?! Finally, how do i integrate over the surface of the pentagon in the double integral?!

Thanks!

2. Feb 27, 2010

### tiny-tim

Hi bon!

Hint: it's not a plane!

(draw a cube, and then trace the line around the cube)

3. Feb 27, 2010

### bon

Ah okay thanks - so do i use divergence theorem to simplify the surface integral over the cube?

How do I use the speed of the flow?

THanks!

4. Feb 27, 2010

### tiny-tim

It might be simplest in this case to just shade in those parts of the faces of the cube that the line crosses, and calculate the flow for each shaded face separately.

5. Feb 28, 2010

### joe:)

Im trying to do a similar problem to this..

As bon says though, I can't see how you use the speed being 0.4m/s here?!

What I would do is dot (-1,-1,1) with n hat for and integrate over two surfaces separately - one with corners (1,1,0), (1,0,0) and (1,1,1)

and then over the other surface..

But how do you use the speed on the vector field?

6. Feb 28, 2010

### joe:)

So summing those two integrals I get..2-1/2 = 1.5

But as you say bon, I'm not sure how to use the speed being 0.4m/s

7. Feb 28, 2010

### tiny-tim

Hi joe:)!

volume per second = length x area per second = speed x area

or, more precisely, = ∫ velocity "dot" normal d(area)

8. Feb 28, 2010

### joe:)

So I need to find a velocity vector in the direction (-1,-1,1) with magnitude 0.4?

So is it 0.4/root3 (-1,-1,1) then do I just dot this with the two normals for the two surfaces and carry out the two double integrals as I did..?

Was I right in getting 1.5? In which case i guess the actual answer should be 4/5root3 - 0.4/2root3 = root3/5? Correct?

THANK YOU :)

9. Feb 28, 2010

### tiny-tim

Hi joe:)!

(have a square-root: √ )
That's right!

(but I haven't checked your actual figures)

10. Feb 28, 2010

### joe:)

Thanks tiny-tim :)
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