Let $f: [0, 1] \to \mathbf{R}$ be a non-negative, continuous function for demonstrating Riemann sums, like $f(x) = x^2$. The Riemann sum approximates the area under the curve with sums of the area of rectangles. There's also the trapezoid method, which is a better numerical approximation but not needed for the sum to converge eventually; Riemann sums will take longer to converge to the correct area, but they'll get there (or close enough) as the mesh size goes to 0.
But those same rectangles and trapezoids don't behave the same way when we try to find the arclength of (the graph of) $f$. To approximate the arclength, we need a polygon along the curve, which corresponds to the tops of trapezoids. If we try to approximate the arc with horizontal line segments (i.e. the tops of rectangles in a Riemann sum), and if we falsely claimed that the sum of their lengths converged to the arclength of $f$, then, as my professor once said, "We'd have the amazing theorem that the arclength of a real function in the plane is equal to the length of its domain." The problem, he said, is that "the error term doesn't vanish fast enough" as the number of line segments increases.
So:
1.) What does it mean for the error to vanish "fast enough"?
2.) Why does it vanish "fast enough" for a Riemann sum to converge to the area under a curve, but not fast enough for tops-of-rectangles in a Riemann sum to converge to the arclength of that curve?
