Evaluate the line integral \(\oint_C P\,dx + Q\,dy\) where \(P = y^2\), \(Q = 2x - 3y\), and \(C\) is the circle \(x^2 + y^2 = 9\), oriented counterclockwise.
Solution
Assume \(C\) is oriented counterclockwise. Green’s theorem gives \[ \oint_C P\,dx+Q\,dy =\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA, \] where \(D\) is the disk \(x^2+y^2\le9\). Here \[ \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=2-2y. \] Therefore \[ \oint_C P\,dx+Q\,dy =2\iint_D dA-2\iint_D y\,dA. \] By symmetry \(\iint_D y\,dA=0\), while \(\operatorname{area}(D)=9\pi\). Hence \[ \oint_C P\,dx+Q\,dy=18\pi. \] With clockwise orientation the value would be \(-18\pi\).