CHAPTER 13 - Basic Integration

Abstract

This chapter discusses the fundamental theory of integration. The fundamental theorem of integral calculus gives a simple way to exactly compute the limit of the sum approximations. In order for sum approximations to tend to an integral, one needs to write them in the form f(a) Δx + f(a + Δx) Δx + f(a + 2Δx) Δx + … + f(b - Δx) Δx, where Δx is the thickness of the slice and f(x) is the variable amount of the slice. This symbolic expression is an important part of the way the formulas are expressed in integration; without the symbolic expression, the more or less obvious approximations could not be computed exactly in a common way. One of the most important algebraic properties of summation, the telescoping sum theorem, plus the differential approximation, or microscope equation for a smooth function gives half of the fundamental theorem of integral calculus. This theorem tells how to find exact symbolic integrals without summing or taking a limit.