Evaluate $CF*dr$ using Green's Theorem. $F(x,y)=y-ln(x2+y2),2tan-1(\frac{y}{x})Cis$ the circle $(x-3)2+(y-5)2=16$ oriented clockwise

$Soby$ Green's theorem, $we$ get:

Where ${}_{D}dA=$ Area $of$ region $D=$ Area $of$ circle

$(x-3{)}^2+(y-2)=4^2$ radius $4$

$=>{}_{D}dA=4^2=16$

$So$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(y-ln(x^2+y^2))dx+2ta{n}^{-1}(\frac{y}{x})dy=-{}_{D}dA$

$=-16$

But

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(y-ln(x^2+y^2))dx+2ta{n}^{-1}(\frac{y}{x})dy=-{\displaystyle \underset{C}{\overset{\phantom{}}{}}}(y-ln(x^2+y^2))c$

$=-(-16)$

$=16$