Numerical method for evaluation triple integrals by using midpoint's rule

Abstract

In this paper, we derive method to find the values of the triple integrals numerically its integrands continuous but have singularity in partial derivatives in the region of the integrals by using Midpoint's rule on the three x,y and dimensions z , and  how to findthe general form of the errors (correction terms) and we will improve the results by using Romberg acceleraion[3],[6] from correction terms that we found it when the number of (l)subintervals that divided interval integral on the exterior dimension z equal to twice the number of subintervals(n) on the interior dimension x and the number of subintervals (m) on the middle dimension y ,that is mean ( h₃=1/2 h₁ , h₁ = h₂ ) when h1 means the distances between the ordinates on the x- axis, h2 means the distances between the ordinates on the y- axis and h3 means the distances between the ordinates on the z-axis and we denote to this method by  Mid³ (h₁,h₂,h₃ ) and we can depend on it to calculate the triple integrals because it gave high accuracy in results by few subintervals .