Does anyone know how to define the two dimensional area cells (Cartesian Product) for the 2D Riemann Sum?
The sum in two dimensions with domain $A$ is defined on Wiki as:
$$ S = \sum_{i=1}^n f(x^*_i, y^*_i) \Delta A_i $$
where $(x^*_i, y^*_i) \in A_i$
I thought of the following, but think it's either over complicated or wrong: $$ I_0 = inf(\{x | (x,y) \in A\}) $$ $$ I_1 = inf(\{y | (x,y) \in A\}) $$ $$ l_{\alpha} = I_0 - sup(\{x | (x,y) \in A\}) $$ $$ l_{\beta} = I_1 - sup(\{y | (x,y) \in A\}) $$ $$ P_0 = \frac {l_{\alpha}} n, P_1 = \frac {l_{\beta}} n $$ $$ A_i = \{(x,y) | (I_0 + P_0 mod(i-1, n) \le x \le I_0 + P_0 mod(i, n)) \land (I_1 + P_1 mod(i-1, n) \le y \le I_1 + P_1 mod(i, n)) \} $$