Assume that $C$ is a negatively oriented, simple, closed curve. Let $R$ be the region enclosed by $C$.

Use the normal form of Green's theorem to rewrite $\oint x^{2} y^{2} d x-5 y^{4}-2 x y d y$ as a double integral.

Choose 1 answer:
(A) $\iint_{R} 20 y^{3}+2 x+2 x y^{2} d A$
(B) $\iint_{R} 2 y\left(1+x^{2}\right) d A$
(C) $\iint_{R}-20 y^{3}-2 x-2 x y^{2} d A$
(D) $\iint_{R}-2 y\left(1+x^{2}\right) d A$
() Green's theorem is not necessarily applicable.

Question

Assume that $C$ is a negatively oriented, simple, closed curve. Let $R$ be the region enclosed by $C$.

Use the normal form of Green's theorem to rewrite $\oint x^{2} y^{2} d x-5 y^{4}-2 x y d y$ as a double integral.

Choose 1 answer:
(A) $\iint_{R} 20 y^{3}+2 x+2 x y^{2} d A$
(B) $\iint_{R} 2 y\left(1+x^{2}\right) d A$
(C) $\iint_{R}-20 y^{3}-2 x-2 x y^{2} d A$
(D) $\iint_{R}-2 y\left(1+x^{2}\right) d A$
() Green's theorem is not necessarily applicable.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Apply Green's theorem to the given line integral. Green's theorem states that for a positively oriented, simple, closed curve $C$ and a region $R$ it encloses, $\oint_C (L dx + M dy) = \iint_R \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA$ where $L$ and $M$ are functions of $x$ and $y$. ‖ ‖ step 2 ⋮ Identify $L$ and $M$ from the integral. Here, $L = x^2y^2$ and $M = -5y^4 - 2xy$. ‖ ‖ step 3 ⋮ Compute the partial derivatives $\frac{\partial M}{\partial x}$ and $\frac{\partial L}{\partial y}$. We find $\frac{\partial M}{\partial x} = -2y$ and $\frac{\partial L}{\partial y} = 2xy^2$. ‖ ‖ step 4 ⋮ Substitute the partial derivatives into the formula from Green's theorem. We get $\iint_R (-2y - 2xy^2) dA$. ‖ ‖ step 5 ⋮ Simplify the double integral expression. The integral becomes $\iint_R -2y(1 + x^2) dA$. ‖ ‖ step 6 ⋮ Since $C$ is negatively oriented, we need to take the negative of the integral we found to correct the orientation. This gives us $\iint_R 2y(1 + x^2) dA$. ‖ ∻[question number] Answer∻ ***B*** ∻Key Concept∻ ⚹Green's theorem and orientation of curve⚹ ∻Explanation∻ ⚹Green's theorem relates a line integral around a simple, closed curve to a double integral over the region it encloses. The orientation of the curve is important; if the curve is negatively oriented, the result of Green's theorem must be negated to match the orientation.⚹◊What is the normal form of Green's theorem and how can it be used to rewrite a line integral as a double integral?⍭ Generate me a similar question◊

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