Implementasi Integral Garis pada Medan Vektor dalam Perhitungan Usaha Gaya

Abstract

Line integrals are one of the fundamental concepts in advanced calculus used to determine the accumulation of a quantity along a specific path within a vector field. This concept has various applications in science, particularly in physics and engineering, especially in calculating the work done by a force acting on an object moving along a certain trajectory. This study aims to examine the application of line integrals in vector fields for calculating the work of a force and to analyze the relationship between a particle’s path and the resulting work. The research method employed is a qualitative approach with mathematical analytical study through theoretical review and calculation of line integrals in two-dimensional vector fields. The analysis process involves defining the vector field function representing the force, parameterizing the particle’s path, and evaluating the line integral to obtain the resulting work. The results show that the work done by a force can be determined through the evaluation of a line integral along the path taken by the particle. Furthermore, the analysis of the vector field indicates that the field is conservative, meaning that the work depends only on the initial and final points of the displacement, not on the shape of the path taken. The visualization of the vector field, particle trajectory graphs, and calculation tables helps provide a clearer understanding of the application of line integrals in analyzing work. Therefore, the concept of line integrals in vector fields can be used as an effective mathematical approach to explain the relationship between force and displacement in a physical system.