Riemann Integral in Two Dimensions

Definition

$\int {\int }_{A}f(x,y)dA={\lim }_{N\to \infty }{\sum }_{i=1}^{N-1}{\sum }_{j=1}^{N-1}f(c_{ij})△x△y$

• For a integral on Region $R=[a,b]\times [c,d]$
• The partitions will be:
  • $\{x_i^{\ast }\},a=x_0<x_1<\cdots <x_N=b$
  • $\{y_i^{\ast }\},c=y_0<\cdots <y_N=d$
• The sample points are the 2d vectors $c_{ij}\in [x_i,x_{i+1}]\times [y_i,y_{i+1}]$