Infinity Times Zero and the Concept of Integration

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).