We are given the field $\textbf{F}(x,y)=(x-y)\textbf{i}+xy\textbf{j}$ and C being $\frac{3}{4}$ of a circle of radius $2$ centered at the origin traversed from $(2,0)$ to $(0,-2)$.
$$\textbf{F}(x,y) =(x-y)\textbf {i}+(xy)\textbf{j}$$
$$C: x^2+y^2=4, x=2\cos(\theta), y=2\sin(\theta), 0 \leq \theta \leq \frac{3\pi}{2}$$
$$dx= -2\sin(\theta), dy=2\cos(\theta)$$
$$\begin{align}\int\limits_C \textbf{F} \dot \ d\textbf{r} & = \int_0^{\frac{3\pi}{2}} \langle 2\cos(\theta)-2\sin(\theta), (4\cos(\theta)\sin(\theta)\rangle \langle -2\sin(\theta), 2\cos(\theta) \rangle \ d\theta \\ & =\int_0^{\frac{3\pi}{2}} 4\cos(\theta)\sin(\theta)+4\sin^2(\theta)+8\cos^2(\theta)\sin(\theta) \ d\theta \\ & =\frac{2}{3}+3\pi\end{align}$$
Did I set up and compute this integral correctly?
Thank you.
