The study addresses the problem of how to present the setting up of definite integrals of real functions like Jbaf(x)dx.In modelling, infinitesimal considerations dating back to the less rigorous times of calculus are still successfully used. On the other hand, students are given a rigorous definition of the definite integral in the calculus course. Different pedagogical solutions to this gap between theory and practice have been presented by teachers. This paper suggests one unified solution based on the error analysis of the termf(x) dx.The starting point here is the interpretation mentioned by some authors that the infinitesimal derivations can be seen as a streamlined or abbreviated version of the use of the rigorous definition of the definite integral. In setting up definite integrals, the termf(x) dxis then an approximation to the quantity to be calculated on the generic subinterval [x,x + dx],where dx is now a positive number, not an infinitesimal. The effect of the corresponding approximation errors in the unabbreviated derivation based on the definition of the definite integral must vanish. This issue is addressed in this paper. General results are given that ensure that the error inf(x)dxis really insignificant guaranteeing thatjbaf(x)dxis the exact value for the quantity to be calculated. The aim of these results is to give rigour and confidence to the student when learning to set up definite integrals.