Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-$$\mathbf { F } = \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { i } + ( x - y ) \mathbf { j } ; C \text { is the rectangle with vertices at } ( 0,0 ) , ( 8,0 ) , ( 8,5 ) , \text { and } ( 0,5 )$$
A) 0
B) 160
C) -160
D) 240

Question

Quizplus · Accepted Answer

Applying Green's Theorem, 
\begin{align*} \oint_C (x^2+y^2)\,dx+(x-y)\,dy &=\iint_R\left(\frac{\partial(x-y)}{\partial x}-\frac{\partial(x^2+y^2)}{\partial y}ight)dA \ &=\iint_R(-2y-2x)dA \ &=\int_0^8\int_0^5(-2y-2x)dydx \ &=\int_0^8(2x-25)dx \ &=\left[x^2-25xight]_0^8 \ &= -	ext{160}\end{align*} 
Therefore, the answer is C.

Using Green's Theorem, Compute the Counterclockwise Circulation of F Around