Meaning
In Korean high school mathematics, 구분구적법 means dividing [a, b] into n equal parts, adding the areas (or volumes) of the pieces, and taking the limit as n → ∞. For y = x² on [0, 1], Σ(k/n)²·(1/n) = (n+1)(2n+1)/(6n²) → 1/3.
History
- Method of exhaustion: refined by Eudoxus (4th century BC) and used by Archimedes, who squeezed the circle between inscribed and circumscribed polygons and showed a parabolic segment is 4/3 of its inscribed triangle.
- Cavalieri's principle (1635): solids whose cross-sections at every height have equal areas have equal volumes, making e.g. the sphere's volume easy.
- Newton and Leibniz: the fundamental theorem of calculus linked area to antiderivatives in the late 17th century.
- Riemann (1854): defined the integral rigorously as a limit of sums over arbitrary partitions.
Link to the definite integral
limn→∞ Σ f(xk)Δx = ∫ab f(x)dx. With finite n an approximation error remains, so experiments analyze it together with measurement error and uncertainty.
References
- Heath, T. L. (Ed.). (1897). The Works of Archimedes. Cambridge University Press.
- Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development. Dover.
- Andersen, K. (1985). Cavalieri's method of indivisibles. Archive for History of Exact Sciences, 31(4), 291-367.
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage.