Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 4 / c / Solution

Subtract the integrands before integrating: ${\mathbf{G}}_2-{\mathbf{G}}_1=(0,y^3,0)$. With the upward vector area element,This extends continuously to the lower edge, soThe individual flux integrals also converge: ${\mathbf{G}}_1\cdot d\mathbf{S}=x(3+2y)dxdy$ gives $I_1=5$, and hence $I_2=7$. Reversing the surface orientation changes both signs and leaves the requested magnitude unchanged.