Green's Theorem:

1. Evaluate the line integral $\displaystyle{\oint_C y\,e^x \, dx + 4\, e^x\, dy }$,  where  $C$  is the positively oriented rectangle with vertices (0, 0), (2, 0), (2, 3) and (0, 3).  solution

2. Evaluate the line integral $\displaystyle{\oint_C \vec{F}\cdot \vec{dr} }$, where $\vec{F}(x,y)=\langle xy, x^2y^3\rangle$ and  $C$  is the positively oriented triangle with vertices (0, 0), (1, 0), and (1, 2).   solution

3. Evaluate the line integral $\displaystyle{\oint_C xy^2\, dx + 4x^2 y \, dy}$, where $C$  is the positively oriented triangle with vertices (0, 0), (2, 2), and (2, 4).    solution

4. Find the work done by the force ${\bf F} = \langle x^2+xy, y+xy^2 \rangle$ in moving an object from (0, 0) to (1, 1) along the curve $y = x^3$, then back to the origin along the line $y=x$.   solution

5. Evaluate the integral $\displaystyle{\oint_C -2y^3\, dx + 2x^3 \, dy }$, where $C$ is the circle $x^2+y^2=9$.   solution