If \[ \overline{F}=(2x^{2}-3z)\mathbf{i}-2xy\mathbf{j}-4x\mathbf{k}, \] then \[ \int_{V}\nabla\cdot\overline{F}\,dV=\_ \] where \(V\) is the closed region bounded by \[ x=0,\quad y=0,\quad z=0,\quad \text{and}\quad x+2y+z=4. \]
Let \( R \) be the planar region bounded by the lines \( x = 0 \), \( y = 0 \) and the curve \( x^2 + y^2 = 4 \) in the first quadrant. Let \( C \) be the boundary of \( R \), oriented counter clockwise. Then, the value of:
\[ \oint_C x(1 - y) \, dx + (x^2 - y^2) \, dy \] is equal to: