Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. ,where C is the triangle with vertices , ,and .

Question

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.  ,where C is the triangle with vertices  ,  ,and  .

Quizplus · Accepted Answer

Using Green's Theorem, we have:
$$\oint_C Mdx+Ndy=\iint_R \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} dA$$
where $M$ and $N$ are the components of the vector field along $x$ and $y$ directions respectively, and $R$ is the region enclosed by the curve $C$.

Let's choose $M=y$ and $N=x^2$ as the components of our vector field. Then $\frac{\partial N}{\partial x}=2x$ and $\frac{\partial M}{\partial y}=1$.

The region $R$ enclosed by $C$ is the triangle with vertices $(2,0)$, $(0,1)$, $(0,0)$. We can see that the region is bound by the $y$ axis on the left, the line $y=x$ on top, and the line $x=0$ at the bottom.

Now we can use Green's Theorem to evaluate the line integral along $C$:
$$\oint_C Mdx+Ndy=\iint_R \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} dA=\iint_R (2x-1) dA$$
Integrating over the region $R$ with the limits $0$ to $1$ for $x$ and $x$ to $0$ for $y$, we get:
$$\int_0^1\int_x^0 (2x-1)dydx=\frac{-1}{2}$$
Therefore, the line integral along $C$ is $\boxed{	extbf{(B) }11eaae29_f431_d523_87fe_9955b7f5932a_TB2067_11}$.

Use Green's Theorem to Evaluate the Line Integral Along the Positively