Evaluate the line integral oint_{C} frac{y}{(x-1)^{2}+4y^{2}} dx+frac{-(x-1)}{(x-1)^{2}+4y^{2}} dy where C is circle defined by x^{2}+y^{2}=27

glasskerfu 2021-01-31 Answered
Evaluate the line integral C y(x1)2+4y2 dx+(x1)(x1)2+4y2 dy where C is circle defined by x2+y2=27
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Expert Answer

mhalmantus
Answered 2021-02-01 Author has 106 answers
Our aim is to apply Green's Theorem. Let us take L(x,y)=y(x1)2+4y2 andM(x,y)=(x1)(x1)2+4y2. It follows thatLy=y(y(x1)2+4y2)
y(y)((x1)2+4y2)y((x1)2+4y2)y((x1)2+4y2)2
=(x1)24y2((x1)2+4y2
Mx=x((x1)(x1)2+4y2)
x(x1)((x1)2+4y2)x((x1)2+4y2)(x1)((x1)2+4y2)2
=(x1)24y2((x1)2+4y2)2Let D be the interior of C, that is, D is a circular disk given by x2+y227. By Green's Theorem, the given line integral can be obtained asCLdx+Mdy=D(MxLy)dxdy

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