Let $P=2x^2 - y^2 $ and $Q=x^2 + y^2$

$\therefore \dfrac {\partial p} {\partial y} = -2$ and $\dfrac {\partial Q} {\partial x} = 2x$

By Green/s Theorem,

$\displaystyle \int_c Pdx + Qdy = \iint_R \left ( \dfrac {\partial Q}{\partial x} - \dfrac {\partial p}{\partial y} \right ) dx \ dy$

$\therefore \displaystyle \int_c (2x^2 - y^2) dx + (x^2 + y^2)dy = \iint_R [2x - (-2y)] dx \ dy$

$= \displaystyle \int^2_0\int^2_0 (2x+2y)dx \ dy$

$= \displaystyle \int^2_0 \left [ 2 \cdot \dfrac {x^2}{2}+2xy \right ]^2 _0 dy$

$= \displaystyle \int^2_0 \Big [( 2^2 + 2\cdot 2y)-0 \Big ] dy$

$= \displaystyle \int^2_ 0 (4+4 y)dy$

$= \left [ 4y + 4\cdot \dfrac {y^2 }{2} \right ]^2_0$

$= \Big [4(2)+ (2^2)\Big ] -\Big [0+0\Big]$

$=16$