How to evaluate integral of (x - y)(dx + dy) with Green's Theorem?

Question

I want to evaluate the integral $\int(x - y)(dx + dy)$ along curve C where C is the semicircular part of $x^2 + y^2 = 4$ above $y = x$ from $(-\sqrt2, -\sqrt2)$ to $(\sqrt2, \sqrt2)$ using Green's Theorem. What is meant by $(x - y)(dx + dy)$? Usually it is in the format $dxdy$.

Answer 1

By Green's theorem,

$$I = \int_c (Pdx+Qdy) = \int\int{\bigg(\frac{\partial Q}{\partial x}} - \frac{\partial P}{\partial y}\bigg)dx\ dy$$

Here $P = Q = x-y$

$Q_x = 1 \ , P_y = -1$, $Q_x - P_y = 2$

$I = \int\int_{s_-}2dxdy$ ( - for clockwise direction)

Now from the graph,

Area of the semicircle (anticlockwise), $$\int\int_{s_+}dxdy = \frac{\pi}{2} r^2 = \frac{\pi}{2}(4) = 2\pi$$

Thus $I = 2\int\int_{s_-}dxdy = -2.2\pi = -4\pi$ (clockwise)