The question is
Calculate the line integral $\int_C v dr$ where $v = (y,-x)$ and $C$ is from $(0,2)$ to $(0,-2)$ along one half of the circle of radius $2$ around the origin.
I know that this is obviously along the circle anti-clockwise between these $2$ points and I have the parametrisation of the circle at $x=2\cos t$ and $y=2\sin t$.
I have tried a few things but I'm not sure what to do with $v$ as it is not in simple function form.
Any help would be great - thanks
Your integral $\int_C v dr$ does not make sense as is, since both $v$ and $r$ are vectors. You have the parameterization of $r$ as $(2\cos t,2\sin t)$. Note that $v$ is $(y,-x)=(2\sin t,-2\cos t)$.
I'm sure the integral you really want is
$$\int_C v\cdot dr$$
Notice the dot product. So find $dr$ from the expression for $r$, take the dot product with $v$, and integrate from $t=0$ to $t=\pi$. You will get a very simple final answer.