Rekonstruksi integrasi numerik metode PIAS segitiga

Abstract

The solution to numerical integration problems generally can be achieved using mesh methods. However, mesh methods, commonly known as trapezoidal, rectangular, and midpoint, only apply to Cartesian coordinates. Therefore, this research develops a mesh method that can be used for numerical integration in polar coordinates, specifically using Triangle shapes. This study also analyzes the errors from the results of the Triangle mesh method and provides examples and visualizations of applying the Triangle mesh method to solve numerical integration problems. The steps of this research are as follows: first, determining the form of the integral in the problem of numerical integration in polar coordinates. Then, the area bounded by the curve is divided into several parts, each approximated by a triangle. Next, a numerical integration formula of the triangle mesh method is created by summing the areas of each triangle. After that, the resulting error of the triangle mesh method is analyzed using the Taylor series. Finally, proving that the results of the triangle mesh method approximate the area bounded by the curve in polar coordinates. From this research, the numerical integration formula for the Triangle mesh method is obtained, the error formula with a second-order approximation degree, and based on the proof of the numerical integration formula for the Triangle mesh method, it is concluded that as the number of triangles approaches infinity, the results of the Triangle mesh method will converge to the exact area bounded by the curve.