According to Green's Theorem, the line integral $\int _ { C } y ^ { 2 } d x + x ^ { 2 } d y$ 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)$2 x + y$
F) $1$
G) $x + 2 y$
H)$2 x + 2 y$

Question

According to Green's Theorem, the line integral $\int _ { C } y ^ { 2 } d x + x ^ { 2 } d y$ 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)$2 x + y$
F) $1$
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 region D it encloses. For the given line integral $\int _ { C } y ^ { 2 } d x + x ^ { 2 } d y$, the function $f(x, y)$ is found by taking the partial derivative of $x^2$ with respect to $x$ and subtracting the partial derivative of $y^2$ with respect to $y$, which results in $2x - 2y$.

According to Green's Theorem, the Line Integral ∫Cy2dx+x2dy\int _ { C } y ^ { 2 } d x + x ^ { 2 } d y∫C​y2dx+x2dy

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