Green's Theorem and Areas

Problem

A bounded simple curve for illustrating Green's theorem.

Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and D be the region bounded by C. If P and Q are functions of (x, y) defined on an open region containing D and have continuous partial derivatives, Green's theorem states that

$$\oint_C P dx + Q dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \,dx\,dy.$$

Use Green's theorem to prove that the area bounded by a simple closed curve can be calculated with the integral

$$\frac{1}{2} \oint_C xdy - ydx.$$

Solution

Let $D$ be the simply-connected domain bounded by a simple closed curve $C$ that is oriented in the positive direction. Green's theorem states that

$$\oint_C P dx + Q dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \,dx\,dy$$

for two continuous functions $P(x, y)$ and $Q(x,y)$ that have continuous first partial derivatives on $D$.

If $P = -y$ and $Q = x$, then

$$\oint_C -y dx + x dy = \iint_D \left(\frac{\partial}{\partial x}(x) - \frac{\partial }{\partial y}(-y) \right) \,dx\,dy$$

$$\oint_C xdy - ydx = \iint_D 2 \,dx\,dy$$

$$\oint_C xdy - ydx = 2A$$

.

Therefore,

$$A = \frac{1}{2} \oint_C xdy - ydx.$$