Arc Length and Surface Area: Are Textbooks on the Same Page?

An element of arc length or an element of surface area can naturally be approximated using either an inscribed object or a tangential object. An approach often followed is to use inscribed objects for arc length and tangential objects for surface area. Why the difference? This paper explores the roads less traveled. Both inscribed and tangential approximations can be used in either situation arc length or surface area. On one hand, the proofs for the inscribed polygonal approach involve deep mathematical analysis, but the approach is more intuitive. On the other hand, the tangential approaches have easier proofs at the expense of some intuition. Also, there is a paradoxical result with a certain polygonal approach for surface area, which will be discussed. In order to appreciate the less traveled approaches, a review of the customary approaches is recommended (see [3] for examples). The less traveled tangential approach to arc length does not require the deep mathematical analysis (like the use of the Mean Value Theorem when using inscribed objects), and hence its proof is considerably less complicated. Let C be defined by y = f(x), for a ≤ x ≤ b, and assume f ′ is continuous. A tangential approximation to C is obtained by creating a partition P of the interval [a, b], say P = {a = x0 < x1 < · · · < xn = b}. Let Ti denote the line segment with endpoints (xi−1, yi−1) and (xi, y∗ i ), where Ti is tangent to C at (xi−1, yi−1) (See Figure 1). The sum of the lengths of these segments is approximately the length of C. So, arc length is defined as