$4\mathrm{.}3$ State and prove the Green's theorem.

$4\mathrm{.}4$ Use Green's Theorem to evaluate the given line integral

$o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}xydy$

where $C$ is the semicircle $x^2+y^2=a^2,y>0$ and the line $y=0,$dots$xa.$