Line integral and greens theorem

[member 428835]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

[itex]y=r\sin\theta[/itex] and [itex]x=r\cos\theta[/itex]. now [itex]\int \vec{F} \cdot d\vec{r}=\int r\cos\theta (\cos\theta dr -r\sin\theta d\theta)[/itex] where [itex]\theta [0,\pi][/itex] and [itex]r [0,1][/itex]
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 [itex]\int \vec{F_x}dx[/itex] since the [itex]dy[/itex] component seems to be zero)

thanks!

 

[LCKurtz]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

[itex]y=r\sin\theta[/itex] and [itex]x=r\cos\theta[/itex]. now [itex]\int \vec{F} \cdot d\vec{r}=\int r\cos\theta (\cos\theta dr -r\sin\theta d\theta)[/itex] where [itex]\theta [0,\pi][/itex] and [itex]r [0,1][/itex]
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 [itex]\int \vec{F_x}dx[/itex] since the [itex]dy[/itex] component seems to be zero)

thanks!

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