López Pouso, Rodrigo2023-07-062023-07-062023López Pouso, R. A roller coaster approach to integration and Peano’s existence theorem. Czech Math J (2023). https://doi.org/10.21136/CMJ.2023.0514-220011-4642http://hdl.handle.net/10347/30840This is a didactic proposal on how to introduce the Newton integral in just three or four sessions in elementary courses. Our motivation for this paper were Talvila’s work on the continuous primitive integral and Koliha’s general approach to the Newton integral. We introduce it independently of any other integration theory, so some basic results require somewhat nonstandard proofs. As an instance, showing that continuous functions on compact intervals are Newton integrable (or, equivalently, that they have primitives) cannot lean on indefinite Riemann integrals. Remarkably, there is a very old proof (without integrals) of a more general result, and it is precisely that of Peano’s existence theorem for continuous nonlinear ODEs, published in 1886. Some elements in Peano’s original proof lack rigor, and that is why his proof has been criticized and revised several times. However, modern proofs are based on integration and do not use Peano’s original ideas. In this note we provide an updated correct version of Peano’s original proof, which obviously contains the proof that continuous functions have primitives, and it is also worthy of remark because it does not use the Ascoli-Arzelà theorem, uniform continuity, or any integration theoryeng© The author(s) 2023.This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/PrimitiveNewton integralPeano’s existence theoremjournal article10.21136/CMJ.2023.0514-221572-9141open access