Example 2.3.6: Line integral along a line parallel to the y-axis

The path is the straight segment $C_1$ from $(3,0,0)$ to $(3,4,0)$. On it $x=3$ and $z=0$ are fixed, so $dx=dz=0$ and $d\ell =dy{a}_y$. Only $A_y$ contributes:

The components $A_x$ and $A_z$ are perpendicular to the path and do no “work”.