$($Chapter $17,$Section $19\mathrm{.}1)$

Consider the intersection $R$ between the two circles $x^2+y^2=2$ and $(x-2{)}^2+y^2=2.$

$($a$)$ Find a $2-$dimensional vector field $F_{?}=(M(x,y),N(x,y))$ such that

$\frac{\text{d}\text{e}\text{l}\text{N}}{\text{d}\text{e}\text{l}\text{x}}-\frac{\text{d}\text{e}\text{l}\text{M}}{\text{d}\text{e}\text{l}\text{y}}=1$

$($b$)$ Using this $F_{?}$ and Green's theorem $($Theorem $19\mathrm{.}1\mathrm{.}1),$ write the area integral ${}_{R}1dA$ as a line integral.

$[$Hint: Any function $y=f(x)$ can be parametrised by $r_t=(t,f(t)).$ This can be used to parametrise the lines$].$

$($c$)$ Using this line integral, find the area of $R.$

$[16]$