Nowadays, geometric modeling is one of the basic components of computer information technology. This fact is due to the simplicity, clarity and ease of use of graphic tools in many areas of human activity. Therefore, further improvement of methods, techniques and algorithms of geometric modeling is an important scientific and technical problem. This applies to the automated design of various products, computer programs for scientific research, modern industrial, medical and household equipment, entertaining games, etc.Bezier curves are quite widely used in current computer graphics systems. This applies to all of the above areas. These lines provide intuitive, well-predicted shaping of various figures in both two-dimensional and three-dimensional geometric modeling, flexible and productive necessary modifications. The Bezier curves have a simple mathematical description and are effectively implemented by computer software and hardware.This article is devoted to the calculating the areas of curved trapezoids bounded by the indicated lines. Such tasks are quite common in engineering practice. For example, these are calculations of cross-sectional areas of various designed structural elements of technical objects. At the same time, the publication discusses in detail the proposed mathematical apparatus, its advantages are compared with existing methods, relevant examples are given, and perspective directions of practical application in modern computer information technologies are defined. The third degree lines are analyzed in the paper as the most widely used. According to the presented approach, the available materials are easily generalized to the Bezier curves of the higher degrees. Thus, the considered mathematical results can be successfully implemented to further improve of various computer tools for modern geometric modeling.