Apply Green's Theorem to evaluate the integral.
-$$\oint _ { C } \left( 7 x + y ^ { 3 } \right) d x + \left( 3 x y ^ { 2 } + 7 y \right) d y$$ C: Any simple closed curve in the plane for which Green's Theorem holds
A) -2
B) 0
C) There is not enough information to evaluate the integral.
D) 2

Question

Quizplus · Accepted Answer

We can apply Green's Theorem, which states that for a region $R$ bounded by a simple closed curve $C$ that is piecewise smooth and oriented counterclockwise, $$\oint_C P\,dx+Q\,dy=\iint_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}ight)\,dA$$ where $P$ and $Q$ are functions whose partial derivatives are continuous on $R$. In this case, we have $P=7x+y^3$ and $Q=3xy^2+7y$. The partial derivatives are $$\frac{\partial Q}{\partial x}=3y^2\qquad	ext{and}\qquad\frac{\partial P}{\partial y}=3y^2$$ which are equal. Therefore, $$\oint_C(7x+y^3)\,dx+(3xy^2+7y)\,dy=\iint_R(3y^2-3y^2)\,dA=0$$ Thus, the answer is $\boxed{	ext{(B) }0}$. Note that any simple closed curve for which Green's Theorem holds will give the same result.

Apply Green's Theorem to Evaluate the Integral ∮C(7x+y3)dx+(3xy2+7y)dy\oint _ { C } \left( 7 x + y ^ { 3 } \right) d x + \left( 3 x y ^ { 2 } + 7 y \right) d y∮C​(7x+y3)dx+(3xy2+7y)dy

Apply Green's Theorem to Evaluate the Integral $\oint _ { C } \left( 7 x + y ^ { 3 } \right) d x + \left( 3 x y ^ { 2 } + 7 y \right) d y$