Given Vector Field:
\[ \vec{A} = (bx+4y^2z)\hat{i} + (x^3\sin z-3y)\hat{j} - (e^x+4\cos(x^2y))\hat{k} \]
For a vector field to be solenoidal,
\[ \nabla \cdot \vec{A}=0 \]
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: