Question

Math

Posted 5 months ago

```
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.
```

Sign in to unlock the answer

Answer from Sia

Posted 5 months ago

Solution by Steps

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.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question