# Calcularing area vector using line integral

Tags:
1. Sep 15, 2015

### SquidgyGuff

1. The problem statement, all variables and given/known data
A closed curve C is described by the following equations in a Cartesian coordinate system:

where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula:

2. The attempt at a solution
I'm mostly having trouble defining what is as a physical quantity. I think it is the distance to an incremented point along the curve such that it the area of the equilateral shape formed by the vectors and that half the integral of that gives the area but i'm suck here:

Than I can evaluate it as:

and this would give a result that is only in the z direction which dimensionally makes sense

2. Sep 15, 2015

### Ray Vickson

If
$$\vec{r} = x(t) \, {\bf i} + y(t) \, {\bf j} + z(t)\, {\bf k}$$
how would you compute $d \vec{r}$ in terms of $dt$?

3. Sep 16, 2015

### SquidgyGuff

My first instinct was just to derive each of them with respect to t like such

Is that right?

4. Sep 16, 2015

### SquidgyGuff

.

5. Sep 16, 2015

### SteamKing

Staff Emeritus
Yes, go ahead and finish the evaluation of the integral ...

6. Sep 16, 2015

### SquidgyGuff

So http://www.sciweavers.org/upload/Tex2Img_1442441349/eqn.png [Broken] (by trig identities)
and so the integral is

Last edited by a moderator: May 7, 2017
7. Sep 16, 2015

### SteamKing

Staff Emeritus
There's a small mistake in your integration. You seem to have omitted integrating the constant (3/8) in your trig identity expression

Last edited by a moderator: May 7, 2017
8. Sep 16, 2015

### SquidgyGuff

I was hoping you wouldn't notice that, I was just too lazy to retype it into LaTex, but yes, it was included in my calculations (I appreciate your thoroughness though!)