Integrals play a fundamental role in various scientific and industrial applications. Their significance extends beyond theoretical mathematics to practical fields such as physics, chemistry, and engineering. One notable application is in volume determination, which is essential for industries designing product containers. This project explores the use of integrals to derive a general expression for the volume of a Pilsen bottle using the Lagrange method. The motivation for this study arose from a simple yet intriguing question: why are certain beverage bottles made of glass despite their fragility? To answer this, we investigate how integral calculus, particularly the disk method and proportionality properties, can be applied to model the bottle’s shape and compute its volume. By integrating mathematical techniques with real-world applications, this research demonstrates the practical utility of integrals in industrial design and production.

Abstract