# Calcularing area vector using line integral

## Homework Statement

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

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

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

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##?

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

Is that right?

SteamKing
Staff Emeritus
Homework Helper
My first instinct was just to derive each of them with respect to t like such

Is that right?
Yes, go ahead and finish the evaluation of the integral ...

Yes, go ahead and finish the evaluation of the integral ...
So http://www.sciweavers.org/upload/Tex2Img_1442441349/eqn.png [Broken] (by trig identities)
and so the integral is

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SteamKing
Staff Emeritus