Evaluate the Line Integral by Applying Green's Theorem: Four Closed Curves

For the following exercises, evaluate the line integrals by applying Green's theorem.

1. $\displaystyle \int_{C} 2xy\,dx + (x+y)\,dy$, where $C$ is the path from $(0,0)$ to $(1,1)$ along the graph of $y=x^{3}$ and from $(1,1)$ to $(0,0)$ along the graph of $y=x$ oriented in the counterclockwise direction.

2. $\displaystyle \int_{C} 2xy\,dx + (x+y)\,dy$, where $C$ is the boundary of the region lying between the graphs of $y=0$ and $y=4-x^{2}$ oriented in the counterclockwise direction.

3. $\displaystyle \int_{C} 2\arctan\!\left(\frac{y}{x}\right)dx + \ln(x^{2}+y^{2})\,dy$, where $C$ is defined by $x=4+2\cos\theta,\ y=4\sin\theta$ oriented in the counterclockwise direction.

4. $\displaystyle \int_{C} \sin x \cos y\,dx + (xy + \cos x \sin y)\,dy$, where $C$ is the boundary of the region lying between the graphs of $y=x$ and $y=\sqrt{x}$ oriented in the counterclockwise direction.