What Are the $x_i^*,y_j^*$ in the Riemann Sum Definition of the Double Integral?

Question

the double integral $$\iint \limits_{[a,b] \times [c,d]} f(x,y) \, dxdy$$ can be represented with a Riemann Sum as $$\lim_{(\Delta x , \Delta y) \to (0,0)} \sum_{i=1}^n\sum_{j=1}^mf(x_i^*,y_j^*)\Delta x \Delta y$$ What are $x_i^*$ and $y_j^*$ in this case?

Answer 1

Depending on the type of Riemann sum you're using (left rectangular, right rectangular, trapezoidal, etc), $(x_i^*, y_i^*)$ is some point in the rectangle $[i\Delta x, (i+1)\Delta x]\times[j\Delta y, (j+1)\Delta y]\subset [a,b]\times [c,d]$.