Let \(R\) be the planar region between the circles \(x^2 + y^2 = 4\) and \((x-1)^2 + y^2 = 1\), and lying in the halfplane \(y \ge 0\). Let \(\partial R\) be its boundary, oriented so that \(R\) is on its left. Either using the definition or by applying theorems of Calculus, compute the line integral \[\int_{\partial R} x\, dy + y\, dx .\]
Solution
The differential form is exact: \[ x\,dy+y\,dx=d(xy). \] Therefore its integral around any closed piecewise smooth curve is zero. Since \(\partial R\) is closed, \[ \int_{\partial R}x\,dy+y\,dx=0. \] Equivalently, Green’s theorem gives the same result because \[ \frac{\partial x}{\partial x}-\frac{\partial y}{\partial y}=1-1=0. \]