@Colin, gradience is really a side effect that makes tangents suitable. The deeper reason is the "convergence" of the approximating shape (convergence from two sides of the curve). Else the jagged steps /rectangles would have landed in deep trouble while approximating the area of the curve or the volumes of shapes. I plan to write another article where non-tangets can approximate the perimeter as well once the convergence criteria is met and why tangents become a special case of that convergence.