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$$
\vec r(t)=\langle \cos t, \sin t \rangle,~~\vec F = \langle \sin t , 0\rangle$$