Calculate the line integral along the curve $\oint_{\gamma} ydx-xdy $ along the curve

Question

$$ \gamma (t) = \cos (t) \; \vec{i} + sin (t) \; \vec{j} $$ with $ \;\; t=0 \ldots\pi $

first using $x$ as a parameter and then $y$ as a parameter.

Answer 1

I don't understand what does "using x and then y as a parameter" mean in this case. Here, you're going to get minus twice the area of the domain determined by $\;\gamma\;$ together with the $\;x\,-$ axis, i.e.: the area of a unit circle (this follows from Green's theorem "closing" the curve):

$$\int_Cydx-xdy=\int_0^\pi\left[\sin t(-\sin t)-\cos t(\cos t)\right]dt=-\int_0^\pi1\,dt=-\pi$$

Using Green's Theorem with the vector field

$$F(x,y)=\left(xy,\,xy\right)\implies\iint_B\left(y-x\right)dA=\int_Cydx-xdy+\int_{C_1}ydx-xdy$$

where $\;C_1: (t,0)\;,\;\;-1\le t\le1\;$ . But

$$\int_{C_1}ydx-xdy=\int_{-1}^1(0\cdot1-t\cdot 0)dt=0$$

and thus the result