Example 2.3.8: Direction of travel

On the $y$-axis $x=z=0$ and $d\ell =dy{a}_y$, so $A\cdot d\ell =3y^{2}dy$. The path $C_3$ starts at $y=3$ and ends at $y=1$, and these are the lower and upper limits:

The answer is negative because we travel in the $-y$ direction while $A_y=3y^2$ is positive. It is a common mistake to write $d\ell =-dy{a}_y$ for this path and keep the limits from $3$ to $1$, which would count the reversal twice and give $+26$. The direction is accounted for once, by the order of the limits.