Let the solid L be given by,
$L=\left\{\left(x,y,z\right)\in R^3∣\:0\le \:z\le \:1-x^2-y^2,\:y\ge \:-x\right\}$
Observe that this solid is limited by the plane $z=0$, the plane $y=-x$ and the paraboloid $z=1-x^2-y^2$.
Let the $F(x,y,z)=(xy,z,x+2y^3)^T$
c) Let $ω$ be the differential form $ω=xy$ $dx$ $+$ $z$ $dy$ $+$ $(x+2y^3)$ $dz$
Calculate $\int _{\partial L}ω.\:$