Mechanistic explanation of integral calculus

• Autores: Richard Sauerheber
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 46, Nº. 3, 2015, págs. 420-425
• Idioma: inglés
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• Resumen

  • The anatomic features of filaments, drawn through graphs of an integral F(x) and its derivative f(x), clarify why integrals automatically calculate area swept out by derivatives. Each miniscule elevation change dFon an integral, as a linear measure, equals the magnitude of square area of a corresponding vertical filament through its derivative. The sum of all dFincrements combine to produce a range ?Fon the integral that equals the exact summed area swept out by the derivative over that domain. The sum of filament areas is symbolized ?f(x)dx, where dxis the width of any filament and f(x) is the ordinal value of the derivative and thus, the intrinsic slope of the integral point dF/dx. dxdisplacement widths, and corresponding dFdisplacement heights, along the integral are not uniform and are determined by the intrinsic slope of the function at each point. Among many methods that demonstrate why integrals calculate area traced out by derivatives, this presents the physical meaning of differentials dxand dF, and how the variation in each along an integral curve explicitly computes area at any point traced by the derivative. This area is the filament width dxtimes its height, the ordinal value of the derivative function f(x), which is the tangent slope dF/dxon the integral. This explains thoroughly but succinctly the precise mechanism of integral calculus.