According to Green's Theorem, the line integral $\int _ { C } x ^ { 2 } d x + y ^ { 2 } d y$ over a positively oriented, piecewise-smooth, simple closed curve C is equal to the double integral $\iint _ { D } f ( x , y ) d A$ over the region D bounded by C. Find the function $f ( x , y )$ .
A) $2 x - y$
B)$x - 2 y$
C)$2 x - 2 y$
D) $0$
E)$1$
F)$2 x + y$
G)$x + 2 y$
H) $2 x + 2 y$

Question

According to Green's Theorem, the line integral $\int _ { C } x ^ { 2 } d x + y ^ { 2 } d y$ over a positively oriented, piecewise-smooth, simple closed curve C is equal to the double integral $\iint _ { D } f ( x , y ) d A$ over the region D bounded by C. Find the function $f ( x , y )$ .
A) $2 x - y$ 
B)$x - 2 y$ 
C)$2 x - 2 y$ 
D) $0$
E)$1$
F)$2 x + y$ 
G)$x + 2 y$ 
H) $2 x + 2 y$

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. According to Green's Theorem, for a vector field $\mathbf{F} = M\mathbf{i} + N\mathbf{j}$, the line integral $\int_C M dx + N dy$ is equal to the double integral $\iint_D \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) dA$. In this case, $M = x^2$ and $N = y^2$, so $\frac{\partial N}{\partial x} = 0$ and $\frac{\partial M}{\partial y} = 0$. Therefore, the function $f(x, y) = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0 - 0 = 0$.

According to Green's Theorem, the Line Integral ∫Cx2dx+y2dy\int _ { C } x ^ { 2 } d x + y ^ { 2 } d y∫C​x2dx+y2dy

According to Green's Theorem, the Line Integral $\int _ { C } x ^ { 2 } d x + y ^ { 2 } d y$