Line integral and greens theorem

[member 428835]

Homework Statement

\int \vec{F} \cdot d\vec{r} where F=<y,0> and \vec{r}=unit circle.

Homework Equations

i'd prefer to do this one without greens theorem (using it is very easy).

The Attempt at a Solution

y=r\sin\theta and x=r\cos\theta. now \int \vec{F} \cdot d\vec{r}=\int r\cos\theta (\cos\theta dr -r\sin\theta d\theta) where \theta [0,\pi] and r [0,1]
but what do i do with my bounds of this single integral? please help! (i used the chain rule with the above substitutions to evaluate \int \vec{F_x}dx since the dy component seems to be zero)

thanks!

 

[LCKurtz]

Homework Statement

\int \vec{F} \cdot d\vec{r} where F=<y,0> and \vec{r}=unit circle.

Homework Equations

i'd prefer to do this one without greens theorem (using it is very easy).

The Attempt at a Solution

y=r\sin\theta and x=r\cos\theta. now \int \vec{F} \cdot d\vec{r}=\int r\cos\theta (\cos\theta dr -r\sin\theta d\theta) where \theta [0,\pi] and r [0,1]
but what do i do with my bounds of this single integral? please help! (i used the chain rule with the above substitutions to evaluate \int \vec{F_x}dx since the dy component seems to be zero)

thanks!

$r=1$ on the circumference of the unit circle. You parameterization should be$$
\vec r(t)=\langle \cos t, \sin t \rangle,~~\vec F = \langle \sin t , 0\rangle$$