Green's Theorem

(requires JavaScript)

1. Let $C$ be the rectangle with vertices $\left(0,0\right)$, $\left(2,0\right)$, $\left(2,3\right)$, and $\left(0,3\right)$. Evaluate ${\displaystyle {\oint }_{C}x{y}^{2}dx+x^{3}dy}$ twice: first directly and then by using Green's Theorem.
   $6$
2. Use Green's Theorem to evaluate the integral ${\displaystyle {\int }_C{e}^{y}dx+2x{e}^{y}dy}$ along the positively oriented square with sides $x=0$, $x=1$, $y=0$, and $y=1$.
   $e-1$
3. Use Green's Theorem to evaluate the integral ${\displaystyle {\int }_C{x}^2{y}^{2}dx+4x{y}^{3}dy}$ along the positively oriented triangle with vertices $\left(0,0\right)$, $\left(1,3\right)$, and $\left(0,3\right)$.
   $\frac{318}{5}$
4. Use Green's Theorem to evaluate the integral ${\displaystyle {\int }_{C}x{e}^{-2x}dx+\left(x^4+2x^2{y}^2\right)dy}$ along the positively oriented boundary of the region between the circles $x^2+y^2=1$ and $x^2+y^2=4$.
   $0$
5. Use Green's Theorem to evaluate ${\displaystyle {\int }_C\mathbf{F}•d\mathbf{r}}$ if $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=⟨y^2\cos <!--FUNCTION\; APPLICATION-->\left(x\right),x^2+2y\sin <!--FUNCTION\; APPLICATION-->\left(x\right)⟩$ and $C$ is the triangle from $\left(0,0\right)$ to $\left(2,6\right)$ to $\left(2,0\right)$ to $\left(0,0\right)$. Make sure to check the orientation of the curve.

6. Let $D$ be a region bounded by a simple closed path $C$ in the $xy\text{-plane}$. Use Green's Theorem to prove that the coordinates of the centroid $\left(\bar{x},\bar{y}\right)$ of $D$ are

   ${\displaystyle \bar{x}=\frac{1}{2A}{\oint }_C{x}^{2}dy}$ and ${\displaystyle \bar{y}=-\frac{1}{2A}{\oint }_C{y}^{2}dx}$,

   where $A$ is the area of $D$.

7. Use a path integral to find the centroid of the triangle with vertices $\left(0,0\right)$, $\left(1,0\right)$, and $\left(0,1\right)$.
   ${\displaystyle \left(\frac{1}{3},\frac{1}{3}\right)}$