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Posted 7 March 2016
INTEGRALS
The integral sign as delimiter
For many authors, the integral sign and the dx in the expression ∫a b x2 +1 dx delimit the integrand
(and of course also provide other information - they are not bare delimiters).
However, many others do not recognize these symbols as delimiters and would write ∫a b ( x2 +1)dx.
There is a strong argument for the latter position. Historically the intuition behind the expression ∫a b f(x)
dx is that it is a sum: the integral sign is an elongated letter "S", and the terms of the sum are the infinitesimal rectangles with height f(x) and width dx. This perception is the motivation for most physical applications of definite integration, so it is reasonable to teach it explicitly and to keep one's notation consistent
with it.
A Garden of Integrals
Frank Burk