id: 06430684
dt: j
an: 2015c.00818
au: Sauerheber, Richard D.
ti: Mechanistic explanation of integral calculus.
so: Int. J. Math. Educ. Sci. Technol. 46, No. 3, 420-425 (2015).
py: 2015
pu: Taylor \& Francis, Abingdon, Oxfordshire
la: EN
cc: I50
ut: integration; integral/derivative pairs; calculus integration mechanism
ci:
li: doi:10.1080/0020739X.2014.956826
ab: Summary: 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 $dF$ on an integral, as a linear measure, equals the
magnitude of square area of a corresponding vertical filament through
its derivative. The sum of all $dF$ increments combine to produce a
range $ΔF$ on the integral that equals the exact summed area swept out
by the derivative over that domain. The sum of filament areas is
symbolized $\int f(x)dx$, where $dx$ is the width of any filament and
$f(x)$ is the ordinal value of the derivative and thus, the intrinsic
slope of the integral point $dF/dx$. $dx$ displacement widths, and
corresponding $dF$ displacement 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 $dx$ and $dF$, 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 $dx$ times its height, the
ordinal value of the derivative function $f(x)$, which is the tangent
slope $dF/dx$ on the integral. This explains thoroughly but succinctly
the precise mechanism of integral calculus.
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