Approximating multiple integrals over non-rectangular compact set using alpha-dense curves

Abstract

In this paper, we develop a method for approximating multiple integrals. The domain of integration $\Omega $ is assumed to be a non-rectangular compact of $\mathbb{R}^{n}$. The main idea is the dimensionality reduction procedure based on the use of parametric $\alpha $-dense curves $\ell_{\alpha }(t)$. First, the region whose measure represents the value of the integral, is densified using new results, by a certain $\alpha $-dense curve of finite length. The multiple integral of a positive continuous function $f$ over $\Omega $ is approximated by a unique single integral corresponding to $\ell_{\alpha }(t)$. Some numerical examples are given.