Section 8–4 Distance as an integral

(Distance in terms of infinitesimals / Integration process / Definite functions)

In this section, the three interesting points discussed are the distance in terms of infinitesimals, integration process, and definite functions.

1. Distance in terms of infinitesimals:

“We use the same idea, and express the distance in terms of infinitesimals. Let us say, ‘In the first second her speed was such and such and from the formula Δs = vΔt…’ (Feynman et al., 1963, section 8.4 Distance as an integral).”

Feynman discusses the inverse problem: how can we deduce the distance if the speed is known? He suggests the idea of expressing the distance in terms of infinitesimals. In a sense, there are some linguistic issues with regard to the words distance and infinitesimals. To be precise, physics teachers should distinguish “distance traveled” and “distance from the origin” (or position). Furthermore, the formula Δs = vΔt as cited in the next sentence is not definitely in terms of infinitesimals. Strictly speaking, this is possible if there is a limiting condition in which Δt approaches zero. Instead of the formula Δs = vΔt, Feynman could state ds = v dt and explain that the infinitesimal ds is the smallest change in displacement (v means the instantaneous velocity).

Some mathematicians may not agree with Feynman in expressing the distance in terms of infinitesimals. The term infinitesimal is a controversial concept and some have explained that this term is dangerous or is no longer used in modern mathematics (Alexander, 2014). They do not agree that infinities or infinitesimals exist in the real world. To the surprise of mathematicians, Robinson (1961) proposes a new way to reintroduce Leibniz’s infinitesimals as a precisely defined mathematical entity. Robinson’s way of using infinitesimals in calculus is known as “non-standard analysis.” There are still some mathematicians criticizing this method of analysis.

2. Integration process:

“This process of adding all these terms together is called integration, and it is the opposite process to differentiation (Feynman et al., 1963, section 8.4 Distance as an integral).”

Feynman’s explanation of integration is rather brief. In essence, he explains that the Δ is replaced by a “d” to remind us that the time is as small as it can be and the addition is written as a sum with a great “s,” ∫ (from the Latin summa). Feynman opines that it is unfortunately just called an integral sign ( ò ) which is only a long S, merely means “the sum of.” However, physics teachers can elaborate that the symbol S may represent the sum of a number of finite quantities, whereas the integral sign ò is always used to represent the summing of an infinite number of “infinitely small quantities.” Alternatively, one may explain geometrically an “integral” as the total area under a curve. Furthermore, we can write the relationship as “distance = ∫ speed (t) dt” or “displacement = ∫ velocity (t) dt.”

Physics teachers should explain why integration is sometimes known as anti-derivative or why would the integral of the velocity bring you back to displacement? We should recall that the area under any curve can be approximated by using a large number of rectangles. If the rectangles are getting thinner and thinner, we can achieve better approximations in determining the correct area. When the limit of rectangles are infinitely thin, we can use the integral symbol as follows: ∫ v(t) dt = ∑_iv(t_i)(t_i+1 − t_i). Importantly, we should remember that the rectangles mainly mean “velocity times time,” which is simply a distance traveled over the little interval, t_i+1 − t_i. By adding together all the distances traveled (or more precisely, displacement) over all those little bits of time, it gives us the net displacement (or change in position).

3. Definite function:

“Every function can be differentiated analytically, i.e., the process can be carried out algebraically, and leads to a definite function (Feynman et al., 1963, section 8.4 Distance as an integral).”

According to Feynman, every function can be differentiated analytically and the process can lead to a definite function. This statement is not true because there are different kinds of functions that cannot be differentiated analytically. First, there are piecewise continuous functions that are not differentiable at points of discontinuity. Second, a function such as sin (1/x) cannot be differentiated at x = 0 because it “oscillates” rapidly and thus, it cannot be defined. Third, a function such as x^1/3 is finite everywhere, but its derivative is infinite at x = 0. In short, Feynman could have said, “most continuous functions can be differentiated analytically.” Nevertheless, there are also exceptions in which non-continuous functions can be differentiated.

Mathematicians or mathematical physicists may find the last paragraph of this section unsatisfactory. Some may add that indefinite integral (ò f(x) dx) is a function, whereas definite integral (ò^b_a f(x) dx) that represents the area under the curve f(x) from x = a to x = b has a definite value. Additionally, Feynman simply mentions that it is not possible to find, analytically, what the integral is for some functions. Therefore, one may clarify that the integral of some functions do not have elementary functions. (An elementary function is a function of one variable which composes a finite number of functions that are algebraic, trigonometric, exponential, logarithmic or constant.) Examples of non-elementary integrals are ò ln(ln x) dx, ò sin x² dx, and ò sin x/x dx.

Questions for discussion:

1. Is it meaningful to define a distance in terms of infinitesimals?

2. How would you explain the integration process geometrically?

3. Can every function be differentiated analytically and resulted in a definite function?

The moral of the lesson: the distance from the origin of an object can be calculated by summing an infinite number of “infinitely thin rectangles” under a curve.

References:

1. Alexander, A. (2014). Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. New York: Farrar, Straus, and Giroux.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Robinson, A. (1961). Non-Standard Analysis. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, ser. A, 64, 432-440.