Performance comparison of approximation area by Monte-Carlo simulation and trapezoidal rule

An area means a kind of largeness of a shape on planes and curved surfaces. It is used when the largeness of a land or an office space are derived as a familiar case. There is a wide range of applications. For example, various kinds of physical quantities, population and number of votes are predicted using area density and there are many methods to derive the area largeness. Simple formulas are used to derive the area of a circle or square, and these are learned in elementary and junior high schools. Also, when finding the area of land, Heron’s formula is used by laying out some triangles. Furthermore, there is a method to obtain it by the definite integral. However, the definite integral cannot be used when a formula of the indefinite integral for the shape is not understood. Namely, it is not suitable in the case of complex function. On the other hand, the Monte-Carlo method and trapezoidal rule are well known as methods of computer processing at deriving approximate area. The performance comparison of area approximation by the Monte-Carlo method and trapezoidal rule is performed in this study, in which both simple and complex shapes are processed. Optimal number of iteration and divide number are implemented to introduce a target accuracy, when using the Monte-Carlo method and trapezoidal rule. Moreover, the iteration number n1 in the Monte-Carlo method and the divided number n2 in trapezoidal rule are derived and the CPU times for both of the methods are introduced in the same precision.