Double Riemann Sum

Definition: Double Riemann Sum

Let $f:R\to \mathbb{R}$ be a real scalar field over some rectangle $R=[a;b]\times [c;d]\subset {\mathbb{R}}^2$.

1. Subdivide the interval $[a;b]$ into $m$ intervals $[x_{i-1};x_i]$ of length $\Delta x=\frac{b-a}{m}$.

2. Subdivide the interval $[c;d]$ into $n$ intervals $[y_{i-1};y_i]$ of length $\Delta x=\frac{d-c}{n}$.

3. These subdivision result in smaller rectangles $R_{ij}=\{(x,y)\mid x_{i-1}\le x\le x_i,y_{j-1}\le y\le y_j\},$ each with area $\Delta A=\Delta x\Delta y$.

• For every choice of $(x_{ij}^{\ast },y_{ij}^{\ast })\in R_{ij}$ we get a double Riemann sum of $f$ over $R$.

$$\sum _{i=1}^m\sum _{j=1}^{n}f(x_{ij}^{\ast },y_{ij}^{\ast })\Delta A$$