91Ó°ÊÓ

In order to apply Green's theorem, we first convert the polar coordinates to Cartesian coordinates. Recall that we can do this using the relationships \(x = rcos(\theta)\) and \(y=rsin(\theta)\). Also, in terms of Cartesian coordinates, an infinitesimal area element \(dA\) in polar coordinates is given by \(rdrd\theta = dxdy\).

Green's theorem is given by \(\oint_{C} P dx + Q dy = \int \int_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA\), where \(P\) and \(Q\) are functions of \(x\) and \(y\), which define a vector field on a region \(D\), and \(C\) is the boundary curve of \(D\). Here we are integrating over the region \(D\) enclosed by curve \(C\), with \(P = 0\) and \(Q = \frac{1}{2}r^{2}\). After converting \(Q\) into cartesian, we find \(Q = \frac{1}{2}(x^{2}+y^{2})\). So \(\int_{C} \frac{1}{2}r^{2} d\theta\) becomes \(\oint_{C} 0 dx + \frac{1}{2}(x^{2}+y^{2}) dy\). Therefore by Green's theorem, \(\oint_{C} 0 dx + \frac{1}{2}(x^{2}+y^{2}) dy = \int \int_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA\) where the left side is equal to the area integral definition.

We note that the double integral represents the area of the region D. To compute this, we must first differentiate \(Q\) and \(P\) with respect to \(x\) and \(y\) respectively. We find that \(\frac{\partial Q}{\partial x} = x\) and \(\frac{\partial P}{\partial y} = 0\). Subtracting these gives us \(x - 0 = x\). Thus, our double integral simplifies to \(\int \int_{D} x dA\). This simplifies even further since the integrand is constant over the region D. Therefore, \(\int \int_{D} x dA = \frac{1}{2}\), which is exactly the formula given for the area of a plane region in polar coordinates.

The area formula \(\frac{1}{2} \int_{C} r^{2} d\theta\) holds true since it corresponds to the double integral over the region D in cartesian coordinates as we demonstrated through Green's theorem. This confirms the integrity of the given line integration formula for finding the area of the region giving confidence for its use in the future.