In this chapter, the second of the processes in the traditional sequence of courses
that comprise the calculus curriculum, “integral calculus”, is examined in terms of its
algebraic foundation. A new paradigm is proposed that views “integration”, also often
designated as “anti- (or inverse) differentiation”, as a form of infinity multiplied by zero.
Such a protocol is demonstrated as being the inverse operation to the previously developed
process of differentiation in Chapter 3. However, unlike the function produced by the
process of differentiation, this inverse is either not unique and needs to be supplemented with
an arbitrary unspecified constant (which is addended to the generated function) or a set of
limits. Along with “sloughing through” several of the techniques associated with such
anti-differentiation, the mathematical underpinning of this inverse operation is introduced.
This is then supplemented by an expansion of the horizon of “what is mathematics?” to
introduce (1) a special (more advanced) function, called the Dirac delta function, which, in an
altogether different manner is also subsumed by the over-arching concept of infinity
multiplied by zero, and (2) the theoretical base from one limited to continuous functions
(called “Riemann integration”) to a larger set that includes selected discontinuous functions
(called “Lebesgue integration”).
Keywords: Anti-Derivatives, Area of Surface of Revolution, Dirac Delta, Direct
Substitutions, Indirect Substitutions, Integral Calculus, Integration By Parts,
Kronecker Delta, Lebesque Integral, Length of Planar Curves, Volume of
Revolution (Slices vs. Shells).
