Question

$$ \begin{array}{l}{\text { Writing to Learn Explain geometrically why it does not work }} \\ {\text { to use short horizontal line segments to approximate the lengths }} \\ {\text { of small arcs when we search for a Riemann sum that leads to the }} \\ {\text { formula for arc length. }}\end{array} $$

Short answer

Geometrically, using horizontal line segments to approximate the lengths of small arcs does not accurately reflect the true length of the curve, as it only considers horizontal movement and not the actual path of the curve. Moreover, the principle of the Riemann integral, by definition, partitions on the x-axis, thereby not appropriate for approximating arc length using short horizontal line segments.

Step-by-step solution

Understanding Arc Length and Riemann Sum

Begin by understanding what arc length and Riemann sums are. The length of a curve, or arc length, is found by creating a close approximation then taking the limit as the number of segments goes to infinity. Riemann sums, a method of approximating the definite integral of a function, uses finite sums to estimate area under a curve.

Role of Horizontal Line Segments

Think about the role of the horizontal line segments while approximating the lengths of small arcs. When we use horizontal line segments, we are basically projecting the curve onto the x-axis. This does not truly reflect the actual length of the curve, which does not lie strictly horizontally.

Arc Length and Geometric Intuition

Now, consider the geometric intuition behind the arc length. If we imagine short line segments along the curve, it is clear that these line segments give a more accurate approximation than horizontal line segments. Each short line segment assumes the change in the curve, considering it in both the x and y directions.

Conceptual Boundaries of Riemann Sum

Ultimately, it's a matter of the definition and conceptual boundaries of the Riemann sum. One fundamental principle of the Riemann integral is that the partition must be a partition of the interval [a, b] - in other words, it has to be along the x-axis. This is why the Riemann sum can't be directly used to approximate arc length using short horizontal lines.