Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 15 / b / i / Solution

On the first segment, $x=z=0$, so both line integrals vanish. On the second, $y=1,z=0$, giving contributions ${\int }_0^{2}3x^{2}dx=8$ and ${\int }_0^{2}2xdx=4$. On the last, $x=2,y=1$, the contributions are ${\int }_0^3-2zdz=-9$ and ${\int }_0^3-6z^{2}dz=-54$. ThereforeThe first also equals $\varphi (B)-\varphi (A)=8-9=-1$, as the fundamental theorem for line integrals predicts.