A Remark on the Change of Variable Theorem for the Riemann Integral

In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function f on its entire domain) can be omitted while the change of variable formula still holds.

Approach to the Formulation of the Variable Change Theorem

This research proposes a didactic strategy to enrich the assimilation processes of the change of variable theorem in solving the definite integral. The theoretical foundations that support it are based on the contributions of social constructivism, problem solving, and treatment of theorems. The practical validation of the strategy is carried out with students of the Higher Technical Level in Applied Mathematics at the Autonomous University of Guerrero.

Abraded Glass Strength: An Ad Hoc Fitting Protocol Based on the Change of Variable Theorem

This work tackles the problem of finding a suitable statistical model to describe relevant glass properties, such as the strength under tensile stress. As known, glass is a brittle material, whose strength is strictly related to the presence of microcracks on its surface. The main issue is that the number of cracks, their size, and orientation are of random nature, and they may even change over time, due to abrasion phenomena. Consequently, glass strength should be statistically treated, but unfortunately none of the known probability distributions properly fit experimental data, when measured on abraded and/or aged glass panes. Owing to these issues, this paper proposes an innovative method to analyze the statistical properties of glass. The method takes advantage of the change of variable theorem and uses an ad-hoc transforming function to properly account for the distortion, on the original probability distribution of the glass strength, induced by the abrasion process. The adopted transforming function is based on micromechanical theory, and it provides an optimal fit of the experimental data.