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Posted 2 months ago

Assume that CC is a negatively oriented, piecewise smooth, closed curve. Let RR be the region enclosed by CC.

Use the circulation form of Green's theorem to rewrite R4x3yx2dA\iint_{R} 4 x^{3} y-x^{2} d A as a line integral.

Choose 1 answer:
(A) Cx2ydx+yx2dy\oint_{C} x^{2} y d x+y x^{2} d y
(B) Cxydx+x2dx\oint_{C}-x y d x+x^{2} d x
(C) Cxy2dxxydy\oint_{C} x y^{2} d x-x y d y
(D) Cy3dx+x2y2dy\oint_{C}-y^{3} d x+x^{2} y^{2} d y
(E) Green's theorem is not necessarily applicable.
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 2
The given double integral is R(4x3yx2)dA\iint_{R} (4x^3y - x^2) dA. To use Green's theorem, we identify PP and QQ such that QxPy=4x3yx2\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 4x^3y - x^2
step 3
Let's assume P(x,y)=f(y)P(x, y) = f(y) and Q(x,y)=4x3yg(x)Q(x, y) = 4x^3y - g(x), where f(y)f(y) and g(x)g(x) are functions to be determined
step 4
Since Qx=12x2yg(x)\frac{\partial Q}{\partial x} = 12x^2y - g'(x) and Py=f(y)\frac{\partial P}{\partial y} = f'(y), we need 12x2yg(x)f(y)=4x3yx212x^2y - g'(x) - f'(y) = 4x^3y - x^2
step 5
Comparing the terms, we see that g(x)=x2g'(x) = x^2 and f(y)=0f'(y) = 0. Integrating, we get g(x)=13x3+C1g(x) = \frac{1}{3}x^3 + C_1 and f(y)=C2f(y) = C_2, where C1C_1 and C2C_2 are constants of integration
step 6
Since the constants of integration do not affect the derivatives, we can choose g(x)=13x3g(x) = \frac{1}{3}x^3 and f(y)=0f(y) = 0. Thus, P(x,y)=0P(x, y) = 0 and Q(x,y)=4x3y13x3Q(x, y) = 4x^3y - \frac{1}{3}x^3
step 7
Applying Green's theorem, the line integral becomes CPdx+Qdy=C0dx+(4x3y13x3)dy\oint_{C} P dx + Q dy = \oint_{C} 0 dx + (4x^3y - \frac{1}{3}x^3) dy
step 8
Simplifying the line integral, we get C13x3dy\oint_{C} -\frac{1}{3}x^3 dy. Since CC is negatively oriented, we need to reverse the sign, resulting in C13x3dy\oint_{C} \frac{1}{3}x^3 dy
step 9
The correct line integral expression that corresponds to the given double integral using Green's theorem is C13x3dy\oint_{C} \frac{1}{3}x^3 dy. None of the answer choices match this expression exactly, but choice (B) is the closest if we consider a missing factor of 13\frac{1}{3} in the x3dyx^3 dy term
1 Answer
Key Concept
Green's theorem in circulation form
Green's theorem relates a double integral over a region to a line integral around the boundary of the region. It requires identifying the correct functions PP and QQ such that their partial derivatives correspond to the given double integral's integrand.

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