Example 2.3.9: Line integral along an oblique line

Along this path both $dx$ and $dy$ are non-zero ($dy=dx$), and in general both $A_{x}dx$ and $A_{y}dy$ must be kept. Here the second happens to vanish because of $z=0$. One could equally use the unit tangent $a_{\ell }=(a_x+a_y)/\sqrt{2}$ and the arc length $d\ell =\sqrt{2}dx$: then $A\cdot a_{\ell }d\ell =(3x^3/\sqrt{2})\sqrt{2}dx=3x^{3}dx$, the same integrand.